LIBRARY 


UNIVERSITY  OF  CALIFORNIA, 


Class 


THE 

PRINCIPLES  AND  METHODS 


OF 


GEOMETRICAL  OPTICS 


ESPECIALLY  AS  APPLIED  TO  THE 
THEORY  OF  OPTICAL  INSTRUMENTS 


BY 


JAMES   P.  C.  SOUTHALL 

.\ 

Professor  of  Physics  in  the  Alabama  Polytechnic  Institute 


OF   THE 

UNIVERSITY 

OF 


Neto  ¥orfc 
THE  MACMILLAN  COMPANY 

LONDON:  MACMILLAN  &  Co.,  LTD. 
IQIO 


COPYRIGHT,  1910 
BY  JAMES   P.  C.  SOUTHALL 


PRESS  or 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER.  PA 


TO 

HENRY  C.  LOME,  ESQ. 

WHOSE    KIND    ENCOURAGEMENT    AND    EFFECTUAL   AID 
WILL   ALWAYS    BE    REMEMBERED 

THIS   VOLUME    IS    GRATEFULLY    INSCRIBED    BY 

THE   AUTHOR 


207831 


PREFACE. 


From  time  to  time,  almost  like  a  voice  crying  in  the  wilderness, 
some  one  is  heard  to  lament  the  apathy  with  which  Geometrical  Optics 
is  regarded  in  this  country  and  in  England;1  although  it  is  sufficient 
merely  to  call  the  roll  of  such  names  as  BARROW,  NEWTON,  COTES, 
SMITH,  BLAIR,  YOUNG,  AIRY,  HAMILTON,  HERSCHEL,  RAYLEIGH,  etc., 
in  order  to  be  reminded  that  this  domain  of  science  was  once  at  any 
rate  within  the  sphere  of  British  influence.  At  present,  however,  it 
can  hardly  be  gainsaid  that  the  great  province  of  applied  optics  is 
almost  exclusively  German  territory ;  so  that  not  only  is  it  a  fact  that 
nearly  all  of  the  extraordinary  developments  of  modern  times  in  both 
the  theory  and  construction  of  optical  instruments  are  of  German 
origin,  but  it  is  equally  true  also  that  until  at  least  quite  recently2 
there  was  actually  no  treatise  on  Optics  in  the  English  language  where 
the  student  could  find,  for  example,  hardly  so  much  as  a  reference  to 
the  remarkable  theories  of  PETZVAL,  SEIDEL  and  ABBE — to  mention 
only  such  names  as  are  inseparably  associated  with  the  theory  of  optical 
imagery.  Partly  with  the  object  of  supplying  this  deficiency,  and 
partly  also  in  the  hope  (if  I  may  venture  to  express  it)  of  rekindling 
among  the  English-speaking  nations  interest  in  a  study  not  only 
abundantly  worthy  for  its  own  sake  and  undeservedly  neglected,  but 
still  capable,  under  good  cultivation,  of  yielding  results  of  far-reaching 

Deferring  to  CZAPSKI'S  Theory  of  Optical  Instruments  (the  first  edition  of  which  was 
published  in  1893)  and  to  the  volume  on  Optics  in  the  ninth  (1895)  edition  of  MUELLER- 
POUILLET'S  Physics,  Professor  SILVANUS  P.  THOMPSON,  in  the  preface  of  his  valuable 
translation  of  Dr.  O.  LUMMER'S  Contributions  to  Photographic  Optics  (London,  1900), 
writes  as  follows: 

"Both  these  works  are  in  German,  and  most  unfortunately  no  translation  of  either  has 
appeared — most  unfortunately,  for  there  is  no  English  work  in  optics  that  is  at  all  com- 
parable to  either  of  these.  I  say  so  deliberately,  in  spite  of  the  admirable  article  by  Lord 
RAYLEIGH  on  'Optics'  in  the  Encyclopaedia  Britannica,  in  spite  of  the  existence  of  those 
excellent  treatises,  HEATH'S  Geometrical  Optics,  and  PRESTON'S  Theory  of  Light.  No 
doubt  such  books  as  HEATH'S  Geometrical  Optics  and  PARKINSON'S  Optics  are  good  in 
.their  way.  They  serve  admirably  to  get  up  the  subject  for  the  Tripos;  but  they  are  far 
too  academic,  and  too  remote  from  the  actual  modern  applications.  In  fact,  the  science 
of  the  best  optical  instrument-makers  is  far  ahead  of  the  science  of  the  text-books.  The 
article  of  Sir  JOHN  HERSCHEL  'On  Light'  in  the  Encyclopedia  Metropolitana  of  1840 
marks  the  culminating  point  of  English  writers  on  optics." 

2H.  DENNIS  TAYLOR'S  A  System  of  Applied  Optics  (London,  1906)  is  a  most  valuable 
work  by  the  inventor  of  the  celebrated  "COOKE"  lenses  for  photography. 


vi  Preface. 

importance  in  nearly  every  field  of  scientific  research,  I  have  prepared 
the  following  work,  wherein  my  endeavour  has  been  to  lay  before  the 
reader  a  connected  exposition  of  the  principles  and  methods  of  Geo- 
metrical Optics,  especially  such  as  are  applicable  to  the  theory  of 
optical  instruments;  and  although  I  am  regretfully  aware  of  many 
shortcomings  in  the  execution  of  this  task,  I  cling  to  the  hope  that 
they  will  be  perhaps  not  so  apparent  to  many  of  my  critics  as  they  are 
to  myself. 

I  have  not  hesitated  to  use,  especially  in  connection  with  the  geo- 
metrical theory  of  optical  imagery  in  Chapters  V  and  VII,  the  elegant 
and  direct  methods  of  the  modern  geometry,  but  these  applications 
are  always  so  simple  and  elementary  that  it  is  hardly  to  be  feared  that 
any  readers  will  be  deterred  thereby. 

In  the  theory  of  optical  imagery  developed  by  GAUSS  with  such  rare 
analytical  skill,  it  is  assumed  that  both  the  aperture  and  the  field  of 
view  of  the  optical  system  of  centered  spherical  surfaces  is  exceedingly 
small,  so  that  all  the  rays  concerned  in  the  production  of  the  image 
are  comprised  within  a  narrow  cylindrical  region  immediately  sur- 
rounding the  optical  axis.  In  the  design  of  telescopes  with  objectives 
of  considerable  diameter,  the  necessity  of  taking  account  of  the  so- 
called  spherical  aberration  due  to  the  increase  of  the  aperture  was  first 
recognized ;  which  led  to  the  well  known  investigations  on  this  subject 
of  EULER,  BESSEL,  AIRY,  GAUSS,  SEIDEL  and  others.  With  the  de- 
velopment of  the  microscope  and  the  birth  and  growth  of  photography, 
new  requirements  had  to  be  filled  in  order  to  portray  parts  of  the 
object  which  were  not  situated  on  the  optical  axis,  so  as  to  correct, 
if  possible,  the  aberrations  due  not  only  to  increase  of  the  aperture 
but  also  to  increase  of  the  field  of  view.  This  difficult  problem,  under- 
taken first  by  PETZVAL  with  only  partial  success,  was  investigated  by 
SEIDEL,  professor  of  mathematics  in  the  University  of  Munich,  in  a 
series  of  papers  contributed  to  the  Astronomische  Nachrichten  in  the 
year  1855;  wherein,  by  an  extension  of  GAUSS'S  methods  so  as  to 
include  in  the  series-developments  the  terms  of  the  next  higher  order, 
elegant  and  entirely  general  formulae  are  derived  in  a  comparatively 
simple  way,  which  enable  one  to  perceive  almost  at  a  glance  how  the 
faults  in  an  image  formed  by  a  centered  system  of  spherical  refracting 
surfaces  are  due  partly  to  the  size  of  the  aperture  and  partly  also  to 
the  extent  of  the  field  of  view.  These  methods  and  theories  are  treated 
at  length  in  Chapter  XII. 

Prism-Spectra  and  the  Chromatic  Aberrations  of  Dioptric  Systems 
are  the  subjects  that  are  included  under  the  head  of  "Colour-Phe- 
nomena" in  Chapter  XIII. 


Preface.  vii 

One  of  the  most  important  divisions  is  Chapter  XIV,  wherein  the 
reader  will  find  a  fairly  complete  treatment  of  ABBE'S  theory  of  the 
limiting  of  the  ray-bundles  by  means  of  perforated  diaphragms  or 
"stops,"  which  has  so  much  to  do  with  the  practical  efficiency  of  an 
actual  optical  instrument. 

Without  entering  more  fully  into  the  contents  of  the  various  chap- 
ters, it  may  be  stated  that  the  work  as  a  whole  is  designed  as  a  general 
introduction  to  the  special  theory  of  optical  instruments  (telescope, 
microscope,  photographic  objective,  etc.,  including  also  the  eye  itself). 
To  discuss  properly  and  fully  each  of  these  types  would  require  a 
separate  and  extensive  volume,  which  I  may  be  induced  to  undertake 
at  some  future  time  as  a  sequel  to  the  present  work. 

A  complete  system  of  notation  which  is  free  from  objection  is  dif- 
ficult to  devise;  and,  in  spite  of  the  pains  I  have  bestowed  on  the 
matter  and  the  importance  which  I  have  attached  to  it,  I  do  not 
doubt  that  fault  will  be  found  not  only  with  the  plan  which  I  have 
adopted  but  with  many  of  the  characters  which  I  have  introduced. 
My  object  has  been  to  make  the  work  convenient  as  a  book  of  reference, 
so  that  the  meaning  of  a  symbol  and  of  the  marks  that  distinguish  it 
would  be  immediately  obvious  as  far  as  possible;  but  in  order  to  aid 
the  reader  still  further  in  this  respect,  the  principal  uses  of  the  letters 
in  both  the  diagrams  and  the  formulae  are  quite  fully  explained  in  an 
appendix  at  the  end  of  the  volume.  In  some  instances  the  same  letter 
or  sign  has  been  employed  deliberately  in  two  or  more  totally  different 
senses,  but  only  where  there  seemed  to  be  no  chance  of  confusion,  and 
because  also  I  have  tried  carefully  to  avoid  resorting  to  strange  and 
uncouth  symbols  which  often  make  a  mathematical  work  appear  to 
be  far  more  difficult  and  uncanny  than  it  really  is. 

The  original  sources  from  which  I  have  borrowed  have  been  given, 
as  far  as  possible,  either  in  the  text  or  in  the  foot-notes.  I  am  espe- 
cially aware  of  how  much  I  have  derived  in  one  way  or  another  from 
Dr.  CZAPSKI'S  epoch-making  book,  Die  Theorie  der  optischen  Instru- 
mente  nach  ABBE,  and  from  Die  Theorie  der  optischen  Instrumente, 
Bd.  I  (Berlin,  1904)  edited  by  Dr.  M.  VON  ROHR  under  the  auspices 
of  CZAPSKI  himself  and  in  collaboration  with  the  staff  of  optical 
engineers  connected  with  the  world-famous  establishment  of  CARL 
ZEISS  in  Jena.  This  latter  work — which  is,  in  fact,  the  offspring  of 
the  former,  and  in  whose  praise  one  might  well  exclaim,  "O  matre 
pulchra  filia  pulchrior!" — is  a  vast  treasury  of  optical  theory  amassed 
by  experts  in  the  various  branches  of  Geometrical  Optics  which  will 
remain  for  many  years  to  come  the  standard  book  of  reference  on  this 
subject. 


viii  Preface. 

I  gladly  take  this  opportunity  of  expressing  my  thanks  to 
Professor  CHARLES  HANCOCK,  of  the  University  of  Virginia,  who 
made  the  drawings  of  the  diagrams,  and  to  my  colleague  Professor 
A.  H.  WILSON  and  my  assistant  Mr.  C.  D.  KILLIBREW  who  have 
helped  me  with  the  proof-reading.  I  esteem  it  a  privilege  to  be  per- 
mitted to  dedicate  the  work  to  HENRY  C.  LOME,  Esq.,  of  Rochester, 
N.  Y. 

JAMES  P.  C.  SOUTHALL. 

AUBURN,  ALA., 

December  i,  1909. 


CONTENTS. 


CHAPTER   I. 

PAGE 

Methods  and  Fundamental  Laws  of  Geometrical  Optics, 

ARTS,  i-io,  §§  1-34 1-32 

Art.  I.     The  Theories  of  Light,  §§1,2 1,2 

Art.  2.     The  Scope  and  Plan  of  Geometrical  Optics,  §§  3,  4 2,  3 

Art.  3.     The  Rectilinear  Propagation  of  Light,  §§  5-8 3-8 

§  6.  HUYGENS'S  Construction  of  the  Wave-Front 4 

§  7.  FRESNEL'S  Extension  of  HUYGENS'S  Method 7 

Art.  4.     Rays  of  Light,  §§  9,  10 8-10 

§  9.  Principle  of  the  Mutual  Independence  of  the  Rays  of  Light.  8 

Art.  5.     The  Behaviour  of  Light  at  the  Surface  of  Separation  of  two 

Isotropic  Media,  §§  1 1-13 10-12 

Art.  6.    The  Laws  of  Reflexion  and  Refraction,  §§  14-22 13-20 

§  15.  The  Laws  of  Reflexion  and  Refraction 13 

§  1 8.  Principle  of  the  Reversibility  of  the  Light-Path 15 

§  19.  The  Laws  of  Reflexion  and  Refraction  as  derived  by  tb? 

Wave-Theory  (HUYGENS'S  Construction) 16 

§  20.  HUYGENS'S  Construction  of  Reflected  Wave 18 

§  21.  HUYGENS'S  Construction  of  Refracted  Wave 18 

Art.  7.     Absolute  Index  of  Refraction  of  an  Optical  Medium,  §§  23-26  20-22 

§  24.  Absolute  Index  of  Refraction 20 

§  26.  Reflexion  Considered  as  a  Special  Case  of  Refraction 22 

Art.  8.     The  Case  of  Total  Reflexion,  §  27 22-25 

Art.  9.     Geometrical  Constructions,  etc.,  §§  28-30 25-28 

§  28.  Construction  of  the  Reflected  Ray 25 

§  29.  Construction  of  the  Refracted  Ray 26 

§  30.  The  Deviation  of  the  Refracted  Ray 27 

Art.  10.     Certain  Theorems  Concerning  the  case  of  so-called  Oblique 

Refraction  (or  Reflexion),  §§  31-34 28-32 

CHAPTER   II. 

Characteristic  Properties  of  Rays  of  Light,  ARTS.  11-15, 

§§35-49 33-50 

Art.  ii.    The  Principle  of  Least  Time  (Law  of  FERMAT),  §§  35-38 33-36 

§38.  The  Optical  Length  of  a  Ray;  and  the  Principle  of  the 

Shortest  Route 35 

ix 


x  Contents. 

PAGE 

Art.  12.     HAMILTON'S  Characteristic  Function,  §§  39-41 36-39 

Art.  13.    The  Law  of  MALUS,  §§  42,  43 39,  40 

Art.  14.     Optical  Images,  §§  44,  45 40-42 

Art.  15.     Character  of  an  Infinitely  Narrow  Bundle  of  Optical  Rays, 

§§  4M9 42-50 

§  46.  Caustic  Surfaces 42 

CHAPTER   III. 

Reflexion  and  Refraction  of  Light-Rays  at  a  Plane 

Surface,  ARTS.  16-20,  §§  50-64 5i~73 

Art.  16.    The  Plane  Mirror,  §§  50,  51 5i~55 

Art.  17.  Trigonometric  Formulae  for  Calculating  the  Path  of  a  Ray 
Refracted  at  a  Plane  Surface.  Imagery  in  the  case  of 
Refraction  of  Paraxial  Rays  at  a  Plane  Surface,  §§  52,  53.  .  55~59 

§  53.  Refraction  of  Paraxial  Rays  at  a  Plane  Surface 57 

Art.  1 8.     Caustic  Surface  in   the  case  of  a  Homocentric   Bundle  of 

Rays  Refracted  at  a  Plane  Surface,  §§  54,  55 59-64 

Art.  19.    Astigmatic  Refraction  of  an  Infinitely  Narrow  Bundle  of 

Rays  at  a  Plane  Surface,  §§  56-62 64-70 

§  57.  The  Meridian  Rays 65 

§  58.  The  Sagittal  Rays 65 

§  59.  Position  of  the  Primary  Image-Point  S',  and  Convergence- 
Ratio  of  Meridian  Rays 67,  68 

§  60.  Position  of  the  Secondary  Image-Point  ~S',  and  Convergence- 
Ratio  of  Sagittal  Rays 68,  69 

§  61.  Astigmatic  Difference 69 

§  62.  Refraction   at  a   Plane   Surface   of  an   Infinitely   Narrow 

Astigmatic  Bundle  of  Incident  Rays. 69 

Art.  20.  Refraction  of  Infinitely  Narrow  Bundle  of  Rays  at  a  Plane: 
Geometrical  Relations  between  Object-Points  and  Image- 
Points,  §§  63,  64 70-73 

§  64.  Construction  of  the  I.  Image-Point 71 

CHAPTER  IV. 

Refraction  through  a  Prism  or  Prism-System, 

ARTS.  21-31,  §§  65-107 74-133 

Art.  21.  Geometrical  Construction  of  the  Path  of  a  Ray  Refracted 
through  a  Prism  in  a  Principal  Section  of  the  Prism, 
§§  65-69 74-80 

Art.  22.    Analytical  Investigation  of   the  Path  of  a  Ray  Refracted 

through  a  Prism  in  a  Principal  Section,  §§  70-73 80-87 

§  71.  Analytical  Investigation  of  the  case  of  Minimum  Deviation  81 


Contents.  xi 

PAGE 

§  72.  Other  Special  Cases 83 

Art.  23.     Path  of  a  Ray  Refracted  across  a  Slab  with  Parallel  Faces, 

§§  74,  75 88-90 

Art.  24.  Refraction,  through  a  Prism,  of  an  Infinitely  Narrow, 
Homocentric  Bundle  of  Incident  Rays,  whose  Chief  Ray 

lies  in  a  Principal  Section  of  the  Prism,  §§  76-81 9O~97 

§76.  Construction  of  the  I.  and  II.  Image-Points  corresponding 

to  a  Homocentric  Object-Point 90 

§  77.  Formulae   for   Calculation   of   the   Positions  on   the   Chief 

Emergent  Ray  of  the  I.  and  II.  Image-Points 92 

§  78  and  §  79.  Convergence-Ratios  of  the  Meridian  and  Sagittal 

Rays 93,  94 

§  80.  The  Astigmatic  Difference 94 

§  81.  Magnitude  of  the  Astigmatic  Difference  in  Certain  Special 

Cases 95 

Art.  25.  Homocentric  Refraction,  through  a  Prism,  of  Narrow,  Homo- 
centric  Bundle  of  Incident  Rays,  with  its  Chief  Ray  lying 

in  a  Principal  Section  of  the  Prism,  §§  82-85 97-105 

§  83.  Analytical  Method 98 

§  84.  Geometrical  Investigation  (according  to  BURMESTER) 99 

Art.  26.     Apparent  Size  of  Image  of  Illuminated  Slit  as  seen  through 

a  Prism,  §  86 105,  106 

Art.  27.    Astigmatic  Refraction  of  Infinitely  Narrow,   Homocentric 

Bundle  of  Incident  Rays  across  a  Slab  with  Plane  Parallel  • 

Faces,  §§  87-90 106-1 1 1 

§  87.  Construction  of  the  I.  and  II.  Image-Points 106 

§  88.  Formulae  for  the  Determination  of  the  Positions  of  the  I. 

and  II.  Image-Points 107 

§  89.  Astigmatic  Difference  in  case  of  a  Slab 108 

Art.  28.  Path  of  a  Ray  Refracted  through  a  system  of  Prisms,  in  the 
case  when  the  Refracting  Edges  of  the  Prisms  are  all 
Parallel,  and  the  Ray  lies  in  a  Principal  Section  Common 

to  all  the  Prisms,  §§  91-94 111-115 

§  92.  Construction  of  the  Path  of  the  Ray 112 

§  93.  Formulae  for  the  Trigonometrical  Calculation  of  the  Path  of 

the  Ray  through  the  System  of  Prisms 113 

§  94.  Condition  that  the  Total  Deviation  shall  be  a  Minimum. .  .  114 

Art.  29.  Refraction,  through  a  System  of  Prisms,  of  an  Infinitely 
Narrow,  Homocentric  Bundle  of  Incident  Rays:  the  Chief 
Ray  thereof  lying  in  a  Principal  Section  Common  to  all  the 

Prisms,  §§  95-99 115-123 

§  95.  Geometrical  Construction  of  the  I.  and  II.  Image-Points. .  .  115 

§  96.  Formulae   for   Calculation   of   the   Positions   on   the   Chief 

Emergent  Ray  of  the  I.  and  II.  Image-Points 117 


xii  Contents. 

PAGE 
§  97.  The  Convergence-Ratios  of  the  Meridian  and  Sagittal  Rays          120 

§  98.  Formula  for  the  Astigmatic  Difference 121 

§  99.  Homocentric  Refraction  through  a  System  of  Prisms 122 

Art.  30.     Path  of  a  Ray  Refracted  Obliquely  through  a  Prism,  §§  100- 

103 123-128 

§  100.  Construction  of  the  Path  of  the  Ray 123 

§  101.  Formulae  for  Calculating  the  Path  of  a  Ray  Refracted 

through  a  Prism  obliquely 124 

§  102.  Deviation    (D)   of   Ray  Obliquely   Refracted   through   a 

Prism 125 

Art.  31.  Homocentric  Refraction  through  a  Prism  of  an  Infinitely 
Narrow,  Homocentric  Bundle  of  Obliquely  Incident  Rays, 
§§  104-107 128-133 

CHAPTER  V. 

Reflexion  and  Refraction  of  Paraxial  Rays  at  a 

Spherical  Surface,  ARTS.  32-38,  §§  108-134 134-173 

Art.  32.     Introduction.     Definitions,  Notations,  etc.,  §§  108,  109.  .  .  .    134-136 
§  109.  Paraxial  Ray 136 

I.  REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL 

MIRROR,  ARTS.  33,  34,  §§  110-117 I37~i47 

£rt.  33.     Conjugate  Axial  Points  in  the  case  of  Reflexion  of  Paraxial 

Rays  at  a  Spherical  Mirror,  §§  110-112 137-142 

§  112.  Focal  Point  and  Focal  Length  of  a  Spherical  Mirror 140 

Art.  34.  Extra-Axial  Conjugate  Points  and  the  Lateral  Magnification 
in  the  case  of  the  Reflexion  of  Paraxial  Rays  at  a  Spherical 

Mirror,  §§  113-117 142-147 

§  113.  Graphical  Method  of  Showing  the  Imagery  by  Paraxial 

Rays 142 

§  116.  The  Lateral  Magnification 145 

II.  REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL 

SURFACE,  ARTS.  35-37,  §§118-129 147-162 

Art.  35.     Conjugate  Axial  Points  in  the  case  of  the  Refraction  of 

Paraxial  Rays  at  a  Spherical  Surface,  §§  118-120 147-152 

§  119.  Construction  of  the  Image-Point  M'  conjugate  to  the  Axial 

Object-Point  M 149 

§  120.  The  Focal  Points  Fand  Ef  of  a  Spherical  Refracting  Surface          150 
Art.  36.     Refraction  of  Paraxial  Rays  at  a  Spherical  Surface.     Extra- 
Axial  Conjugate  Points.     Conjugate  Planes.     The  Focal 

Planes  and  the  Focal  Lengths,  §§121-124 153-158 

§  122.  The  Construction  of  the  Image-Point  Q'  Corresponding  to 


Contents.  xiii 

PAGB 

the  Extra-Axial  Object-Point  Q 153 

§  123.  The  Focal  Planes  of  a  Spherical  Refracting  Surface 154 

§  124.  The  Focal  Lengths/ and  e'  of  a  Spherical  Refracting  Surface  155 
Art.  37.     The  Image-Equations  in  the  case  of  the  Refraction  of  Par- 
axial  Rays  at  a  Spherical  Surface,  §§  125-129 158-162 

§  125.  The  Abscissa-Equation  in  Terms  of  the  Constants  n,  n' 

and  r 158 

§  126.  The  so-called  Zero-Invariant 159 

§  127.  The  Lateral  Magnification 160 

§  128.  The  Image-Equations  in  Terms  of  the  Focal  Lengths/,  e'. .  161 

III.  SUPPLEMENT:  CONTAINING  CERTAIN  SIMPLE  APPLI- 
CATIONS OF  THE  METHODS  OF  PROJECTIVE 

GEOMETRY,  ART.  38,  §§  130-134 162-173 

Art.  38.     Central  Collineation  of  Two  Plane-Fields,  §§  130-134 162-173 

§  131.  Projective  Relation  of  Two  Collinear  Plane-Fields 163 

§  132.  Geometrical  Constructions 165 

§  133.  The  Invariant  in  the  Case  of  Central  Collineation 168 

§  134.  The  Characteristic  Equation  of  Central  Collineation 170 

CHAPTER  VI. 

Refraction  of  Paraxial  Rays  through  a  Thin  Lens  or  through 

a  System  of  Thin  Lenses,  ARTS.  39-42,  §§  135-152 174-197 

Art.  39.     Refraction  of  Paraxial  Rays  through  a  Centered  System  of 

Spherical  Surfaces,  §§  135-139 174-179 

§  135.  Centered  System  of  Spherical  Surfaces 174 

§  138.  The  Lateral  Magnification  Y 178 

§  139.  The  Principal  Points  of  a  Centered  System  of  Spherical 

Surfaces 178 

Art.  40.     Types  of  Lenses.     Optical  Centre  of  Lens,  §§  140-142 179-182 

§  142.  Optical  Centre  of  Lens 181 

Art.  41.     Formulae  for  the  Refraction  of  Paraxial  Rays  through  an 

Infinitely  Thin  Lens,  §§  143-149 182-191 

§  144.  Conjugate  Axial  Points  in  the  case  of  the  Refraction  of  Par- 
axial  Rays  through  an  Infinitely  Thin  Lens 183 

§  145.  The  Focal  Points  of  an  Infinitely  Thin  Lens 184 

§  146.  The  Focal  Lengths/  and  e'  of  an  Infinitely  Thin  Lens 186 

§  147.  Lateral  Magnification  of  an  Infinitely  Thin  Lens 187 

§  148.  Construction  of  the  Image  Formed  by  the  Refraction  of 

Paraxial  Rays  through  an  Infinitely  Thin  Lens 187 

§  149.  Refraction  of  Paraxial  Rays  through  a  Combination  of 

Infinitely  Thin  Lenses 190 

Art.  42.     COTES'S  Formula  for  the  "Apparent  Distance"  of  an  Object 

viewed  through  any  number  of  Thin  Lenses,  §§  150-152. .  I9~>I97 


xiv  Contents. 

PAGB 
CHAPTER  VII. 

The  Geometrical  Theory  of  Optical  Imagery, 

ARTS.  43-52r§§  153-187 198-262 

I.   INTRODUCTION,  ART.  43,  §§  153-156 198-201 

Art.  43.    ABBE'S  Theory  of  Optical  Imagery,  §§  153-156 198-201 

II.  THE  THEORY  OF  COLLINEATION,  WITH  SPECIAL  REFER- 
ENCE TO  ITS  APPLICATIONS  TO  GEOMETRICAL 

OPTICS,  ARTS.  44-47,  §§  157-171 201-217 

Art.  44.    Two  Collinear  Plane-Fields,  §§  157-161 201^206 

§  157.     Definitions 201 

§  158.     Projective  Relation  of  Two  Collinear  Plane-Fields 202 

§  159.  The  so-called  "Fluent"  Points  of  Conjugate  Rays 203 

§  160.  The  so-called  "Flucht"  Lines  (or  Focal  Lines)  of  Conjugate 

Planes 204 

§  161.  Affinity  of  Two  Plane-Fields 206 

Art.  45.    Two  Collinear  Space-Systems,  §§  162-165 206-210 

§  164.  The  so-called  "Flucht"  Planes,  or  Focal  Planes,  of  Two 

Collinear  Space-Systems 208 

§  165.  Affinity-Relation  between  Object-Space  and  Image-Space.  209 

Art.  46.     Geometrical   Characteristics  of  Object-Space   and    Image- 
Space,  §§  166-168 „ 210-212 

§  166.  Conjugate  Planes 210 

§  167.  The  Focal  Points  and  the  Principal  Axes  of  the  Object- 
Space  and  the  Image-Space 211 

§  168.  Axes  of  Co-ordinates 212 

Art.  47.     Metric  Relations,  §§  169-171 213-217 

§  169.  Relations  between  Conjugate  Abscissae 213 

§  170.  The  Lateral  Magnifications 214 

§  171.  The  Image-Equations 216 

III.  COLLINEAR  OPTICAL  SYSTEMS,  ARTS.  48-52, 

§§  172-187 218-262 

Art.  48.     Characteristics  of  Optical  Imagery,  §§  172-176 218-229 

§  172.  Signs  of  the  Image-Constants  a,  b  and  c 218 

§  174.  Symmetry  around  the  Principal  Axes 221 

§  175.  The  Different  Types  of  Optical  Imagery 223 

Art.  49.    The  Focal  Lengths,  Magnification-Ratios,  Cardinal  Points, 

etc.,  §§  177-182 229-242 

§  177.  Analytical  Investigation  of  the  Relation  between  a  Pair  of 

Conjugate  Rays 229 

§  178.  The  Focal  Lengths  /  and  e' 232 


Contents.  xv 

PAGE 

§  179.  The  Magnification-Ratios  and  their  Relations  to  one  an- 
other   234 

§  1 80,  The  Cardinal  Points  of  an  Optical  System 236 

§  181.  The  Image-Equations  referred  to  a  Pair  of  Conjugate  Axial 

Points 239 

§  182.  Geometrical  Constructions  of  Conjugate  Points  of  an  Op- 
tical System 241 

Art.  50.     Telescopic  Imagery,  §§  183,  184 243-245 

§  183.  The  Image-Equations  in  the  Case  of  Telescopic  Imagery. .  243 

§  184.  Characteristics  of  Telescopic  Imagery 244 

Art.  51.     Combination  of  Two  Optical  Systems,  §§  185,  186 245-255 

§  185.  The  Problem  in  General 245 

§  1 86.  Special  Cases  of  the  Combination  of  Two  Optical  Systems          251 
Art.  52.     General  Formulae  for  the  Determination  of  the  Focal  Points 

and  Focal  Lengths  of  a  Compound  Optical  System,  §  187. .   255-262 

CHAPTER  VIII. 

Ideal  Imagery  by  Paraxial  Rays.    Lenses  and  Lens- 
Systems,  ARTS.  53-58,  §§  188-204 263-286 

Art.  53.     Introduction,  §§  188,  189 263,  264 

Art.  54.    The  Focal  Lengths  of  a  Centered  System  of  Spherical  Sur- 
faces, §§  190-193 264-267 

§  193.  Ratio  of  the  Focal  Lengths  /  and  e' 266 

Art.  55.     Several  Important  Formulae  for  the  case  of  the  Refraction  of 
Paraxial  Rays  through  a  Centered  System  of  Spherical 

Surfaces,  §§  194-196 267-273 

§  194.  ROBERT  SMITH'S  Law 267 

§  195.  Formulae  of  L.  SEIDEL 269 

LENSES  AND  LENS-SYSTEMS,  ARTS.  56-58,  §§  197-204. .  273-286 

Art.  56.     Thick  Lenses,  §§  197-199 273-283 

§  199.  Character  of  the  Different  Forms  of  Lenses 276 

Art.  57.    Thin  Lenses,  §§  200,  201 283-284 

§  201.  Infinitely  Thin  Lenses 284 

Art.  58.     Lens-Systems,  §§  202-204 284-286 

CHAPTER  IX. 

Exact  Methods  of  Tracing  the  Path  of  a  Ray  Refracted 

at  a  Spherical  Surface,  ARTS.  59-65,  §§205-220 287-315 

Art.  59.     Introduction,  §  205 287,  288 

Art.  60.     Geometrical  Method  of  Investigating  the  Path  of  a  Ray 

Refracted  at  a  Spherical  Surface,  §§  206-208 288-294 

§  206.  Construction  of  the  Refracted  Ray 288 

§  207.  "Aplanatic"  Pair  of  Points  of  a  Spherical  Refracting  Surface          290 
§  208.  Spherical  Aberration 292 


xvi  Contents. 

PAGE 

TRIGONOMETRIC  COMPUTATION  OF  THE  PATH  OF  A  RAY  OF  FINITE  INCLINA- 
TION TO  THE  Axis,  REFRACTED  AT  A  SINGLE  SPHERICAL 

SURFACE,  ARTS.  61-65,  §§  209-220 294-315 

CASE  I.  WHEN  THE  PATH  OF  THE  RAY  LIES  IN  A  PRINCIPAL 
SECTION  OF  THE  SPHERICAL  REFRACTING  SURFACE, 

ARTS.  61-63,  §§  209-212 294-304 

Art.  61.  The  Ray-Parameters,  and  the  Relations  between  them, 

§§  209,  210 294-297 

Art.  62.  Trigonometric  Computation  of  the  Path  of  the  Refracted 

Ray,  §  211 298-302 

Art.  63.  Formulae  for  Finding  the  Point  of  Intersection  and  the 
Inclination  to  each  other  of  a  Pair  of  Refracted  Rays  lying 
in  the  Plane  of  a  Principal  Section  of  the  Spherical  Refract- 
ing Surface,  §  212 302-304 

CASE  II.  WHEN  THE  PATH  OF  THE  RAY  DOES  NOT  LIE  IN 
A  PRINCIPAL  SECTION  OF  THE  SPHERICAL  REFRACTING 

SURFACE,  ARTS.  64,  65,  §§  213-220 304-315 

Art.  64.     Parameters  of  Oblique  Ray,  §§  213-215 304-310 

§  214.  Method  of  A.  KERBER 305 

§  215.  Method  of  L.  SEIDEL 307 

Art.  65.  Trigonometric  Computation  of  Path  of  Ray  Refracted  Ob- 
liquely at  a  Spherical  Surface,  §§  216-220 310-315 

§  216.  The  Refraction-Formulae  of  A.  KERBER 310 

§  219.  The  Refraction-Formulae  of  L.  SEIDEL 313 

CHAPTER  X. 

Trigonometric  Formulae  for  Calculating  the  Path  of 
a  Ray  through  a  Centered  System  of  Spherical 
Refracting  Surfaces,  ARTS.  66-69,  §§  221-229 316-330 

CASE  I.  WHEN  THE  RAY  LIES  IN  THE  PLANE  OF  A 

PRINCIPAL  SECTION,  ARTS.  66,  67,  §§  221-224 316-321 

Art.  66.  Calculation-Scheme  for  the  Path  of  a  Ray  lying  in  the  Plane 
of  a  Principal  Section  of  a  Centered  System  of  Spherical 
Refracting  Surfaces,  §§  221-223 316-318 

Art.  67.     Numerical  Illustration,  §  224 318-321 

CASE  II.  WHEN  THE  PATH  OF  THE  RAY  DOES  NOT  LIE 

IN  THE  PLANE  OF  A  PRINCIPAL  SECTION  OF  THE 

CENTERED  SYSTEM  OF  SPHERICAL  REFRACTING 

SURFACES,  ARTS.  68,  69,  §§  225-229 322-330 

Art.  68.  Trigonometric  Formulae  of  A.  KERBER  for  Calculating  the 
Path  of  an  Oblique  Ray  through  a  Centered  System  of 
Spherical  Refracting  Surfaces,  §§  225,  226 322-325 


Contents.  xvii 

PAGE 

§  226.  The  Initial  Values 323 

Art.  69.     Trigonometric  Formulae  of  L.  SEIDEL  for  Calculating  the 
Path  of  an  Oblique  Ray  through  a  Centered  System  of 

Spherical  Refracting  Surfaces,  §§  227-229 325-330 

§  228.  SEIDEL'S  "Control"  Formulae 328 

§  229.  The  Initial  Values 329 

CHAPTER  XL 

General  Case  of  the  Refraction  of  an  Infinitely  Narrow  Bundle 
of  Rays  through  an  Optical  System.    Astigmatism, 

ARTS.  70-78,  §§  230-251 331-366 

Art.  70.     General  Characteristics  of  a  Narrow  Bundle  of  Rays  Re- 
fracted at  a  Spherical  Surface,  §§  230-232 331-336 

§  230.  Meridian  and  Sagittal  Rays 331 

§231.  Different  Degrees  of  Convergence  of  the  Meridian  and 

Sagittal  Rays 333 

§  232.  The  Image-Lines 334 

Art.  71.    The  Meridian  Rays,  §§  233-237 336~343 

§  233.  Relation  between  Object-Point  5  and  the  I.  Image-Point  S'          336 

§  234.  Centre  of  Perspective  K 339 

§  235.  The  Focal  Points  /  and  1'  of  the  Meridian  Rays 340 

§  236.  Formula  for  Calculating  the  Position  of  the  I.  Image-Point 
S'  corresponding  to  an  Object-Point  S  on  a  given  incident 

chief  ray  u .  .  342 

§  237.  Convergence-Ratio  of  Meridian  Rays 342 

Art.  72.    The  Sagittal  Rays,  §§  238-241 343~345 

§  238.  Relation  between  the  Object- Point  ~S  and  the  II.  Image- 
Point  S' L. ._. 343 

§  239.  The  Focal  Points  /,  /'  of  the  Sagittal  Rays 343 

§  240.  Formula  for  Calculating  the  Position  of  the  II.  Image- 
Point  "5'  corresponding  to  an  Object-Point  S  on  a  given 

chief  incident  ray  u 344 

§  241.  Convergence-Ratio  of  the  Sagittal  Rays 345 

Art.  73.    The  Astigmatic  Difference,  and  the  Measure  of  the  Astigma- 
tism, §  242 345-347 

Art.  74.     Historical  Note,  concerning  Astigmatism,  §  243 347,  348 

Art.  75.     Inquiry  as  to  the  Nature  and  Position  of  the  Image  of  an 
Extended  Object  formed  by  Narrow  Astigmatic  Bundles 

of  Rays,  §  244 349~35l 

Art.  76.     Collinear  Relations  in  the  case  of  the  Refraction  of  a  Narrow 

Bundle  of  Rays  at  a  Spherical  Surface,  §§  245,  246 351-356 

§  245.  The  Principal  Axes  of  the  Two  Pairs  of  Collinear  Plane 

Systems 351 


xviii  Contents. 

PAGE 

§  246.  The  Focal  Lengths 354 

Art.  77.     Refraction  of  Narrow  Bundle  of  Rays  through  a  Centered 

System  of  Spherical  Refracting  Surfaces,  §§  247,  248 356-360 

§  247.  Formulae  for  Calculating  the  Astigmatism  of  the  Bundle 

of  Emergent  Rays 356 

§  248.  Collinear  Relations 358 

Art.  78.     Special  Cases,  §§  249-251 360-366 

§  249.  The  Special  Case  of  the  Refraction  of  a  Narrow  Bundle  of 

Rays  at  a  Plane  Surface 360 

§  250.  Reflexion  at  a  Spherical  Mirror  Treated  as  a  Special  Case 

of  Refraction  at  a  Spherical  Surface 361 

§  251.  Astigmatism  of  an  Infinitely  Thin  Lens 363 

CHAPTER  XII. 

The  Theory  of  Spherical  Aberrations,  ARTS.  79-104, 

§§  252-326 367-473 

I.  INTRODUCTION,  ARTS.  79,  80,  §§  252-259 367-376 

Art.  79.     Practical  Images,  §  252 367-369 

Art.  80.    The  so-called  SEIDEL  Imagery,  §§  253-259 369-3/6 

§  254.  Order  of  the  Image,  according  to  PETZVAL 370 

§  255.  Parameters  of  Object-Ray  and  Image-Ray,  according  to 

L.  SEIDEL. 371 

§  256.  The  Correction-Terms  or  Aberrations  of  the  3rd  Order. . .  373 

§  257.  Planes  of  the  Pupils  of  the  Optical  System 374 

§  258.  Chief  Ray  of  Bundle 375 

§  259.  Relative  Importance  of  the  Terms  of  the  Series-Develop- 
ments of  the  Aberrations  of  the  3rd  Order 375 

II.  THE  SPHERICAL  ABERRATION  IN  THE  CASE  WHEN 
THE  OBJECT-POINT  LIES  ON  THE  OPTICAL  Axis, 

ARTS.  81-85,  §§  260-275 376-400 

Art.  81.     Character  of  a  Bundle  of  Refracted  Rays  Emanating  Origi- 
nally from  a  Point  on  the  Optical  Axis,  §§  260-262 376-380 

§  260.  Longitudinal  Aberration,  or  Aberration  along  the  Optical 

Axis .- 376 

§  261.  Least  Circle  of  Aberration 378 

§  262.  The  so-called  Lateral  Aberration 379 

Art.  82.     Development  of  the  Formula  for  the  Spherical  Aberration  of 

a  Direct  Bundle  of  Rays,  §§  263-266 380-386 

§  266.  ABBE'S  Measure  of  the  "Indistinctness"  of  the  Image 385 

Art.  83.     Spherical  Aberration  of  Direct  Bundle  of  Rays  in  Special 

Cases,  §§  267-272 386-394 


Contents.  xix 

PAGE 

§  267.  Case  of  a  Single  Spherical  Refracting  Surface 386 

§  268.  Case  of  an  Infinitely  Thin  Lens 387 

§  271.  Case  of  a  System  of  Two  or  More  Thin  Lenses 392 

Art.  84.     Numerical   Illustration  of  Method  of  Using  Formulae  for 

Calculation  of  Spherical  Aberration,  §  273 394~397 

Art.  85.     Concerning  the  Terms  of  the  Higher  Orders  in  the  Series- 
Development  of  the  Longitudinal  Aberration,  §§  274,  275. .  397-400 
§  275.  The  Aberration  Curve 398 

III.  THE  SINE-CONDITION.     (OPTICAL  SYSTEM  OF  WIDE 
APERTURE  AND  SMALL  FIELD  OF  VISION.), 

ARTS.  86-90,  §§  276-284 400-415 

Art.  86.     Derivation  and  Meaning  of  the  Sine-Condition,  §§  276-278.  400-407 

§  278.  Other  Proofs  of  the  Sine-Law 405 

Art.  87.     Aplanatism,  §  279 407,  408 

§  279.  Aplanatic  Points 407 

Art.  88.    The  Sine-Condition  in  the  Focal  Planes,  §  280 408,  409 

Art.  89.     Only  One  Pair  of  Aplanatic  Points  Possible,  §  281 409-412 

Art.  90.     Development  of  the  Formula  for  the  Sine-Condition  on  the 
Assumption  that  the  Slope-Angles  are  Comparatively  Small, 

§§  282-284 412-415 

IV.  ORTHOSCOPY.    CONDITION  THAT  THE  IMAGE  SHALL  BE 

FREE  FROM  DISTORTION,  ARTS.  91-94,  §§  285-294 415-429 

Art.  91.     Distortion  of  the  Image  of  an  Extended  Object  Formed  by 

Narrow  Bundles  of  Rays,  §§  285-287 415-418 

§  286.  Image-Points  regarded  as  lying  on  the  Chief  Rays 416 

§  287.  Measure  of  the  Distortion 417 

Art.  92.     Conditions  of  Orthoscopy,  §§  288-291 418-422 

§  288.  General  Case:  When  the  Centres  of  the  Pupils  are  Affected 

with  Aberrations 418 

§  289.  Case  when  the  Pupil-Centres  are  without  Aberration 420 

§  290.  The  Two  Typical  Kinds  of  Distortion 421 

§  291.  Distortion  when  the  Pupil-Centres  are  the  Pair  of  Aplanatic 

Points  of  the  System 421 

Art.  93.     Development  of  the  Approximate  Formula  for  the  Distortion 
Aberration  in  case  the  Slope-Angles  of  the  Chief  Rays  are 

Small,  §  292 422-427 

Art.  94.     The  Distortion-Aberration  in  Special  Cases,  §§  293,  294. . . .  427-429 

§  293.  Case  of  a  Single  Spherical  Refracting  Surface 427 

§  294.  Case  of  an  Infinitely  Thin  Lens 428 


xx  Contents. 

PAGE 

V.  ASTIGMATISM  AND  CURVATURE  OF  THE  IMAGE, 

ARTS.  95-98,  §§  295-306 429-444 

Art.  95.    The  Primary  and  Secondary  Image-Surfaces,  §  295 429,  430 

Art.  96.     The  Aberration-Lines,  in  a  Plane  Perpendicular  to  the  Axis, 

of  the  Meridian  and  Sagittal  Rays,  §§  296-298 430-434 

§  297.  Case  when  the  Slope-Angles  of  the  Chief  Rays  are  Small .  .  432 

Art.  97.     Development  of  the  Formulae  for  the  Curvatures  i/R',  i/^', 

§§  299-304 434-441 

§  299.  The  Invariants  of  Astigmatic  Refraction 434 

§  300.  Developments  of  i/s  and  i/s  in  a  series  of  powers  of  <(> 435 

§  301.  The  Expressions  for  the  Co-efficients  B,  B  and  Bf,  ~B' 437 

§  303.  Curvature  of  the  Stigmatic  Image 439 

§  304.  Formulae  for  the  Magnitudes  of  the  Aberration-Lines 440 

Art.  98.     Special  Cases,  §§  305,  306 442-444 

§  305.  Case  of  a  Single  Spherical  Refracting  Surface 442 

§  306.  Case  of  an  Infinitely  Thin  Lens 443 

VI.  ABERRATIONS  IN  THE  CASE  OF  IMAGERY  BY  BUNDLES  OF 

RAYS  OF  FINITE  SLOPES  AND  OF  SMALL,  FINITE 

APERTURES,  ARTS.  99-101,  §§  307-315 444~456 

Art.  99.    Coma,  §§  307,  308 444~448 

§  307.  The  Coma  Aberrations  in  General 444 

§  308.  The  Lack  of  Symmetry  of  a  Pencil  of  Meridian  Rays  of 

Finite  Aperture 447 

Art.  loo.     Formulae  for  the  Comatic  Aberration-Lines,  §§  309-313. . . .  448-455 
§  309.  Invariant-Method  of  ABBE 448 

Art.  101.     Special  Cases,  §§  314,  315 455,  456 

§  314.  Case  of  Single  Spherical  Refracting  Surface 455 

§  315.  Case  of  Infinitely  Thin  Lens 456 

VII.  SEIDEL'S  THEORY  OF  THE  SPHERICAL  ABERRATIONS  OF 

THE  THIRD  ORDER,  ARTS.  102-104,  §§  316-326 456-473 

Art.  102.     Development  of  SEIDEL'S  Formulae  for  the  y-  and  3-Aberra- 

tions,  §§  316-322 456-468 

§  316.  GAUSsian  Parameters  of  Incident  and  Refracted  Rays.  . . .  456 

§  317.  Approximate  Values  of  the  GAUSsian  Parameters,  and  the 

Correction-Terms  of  the  3rd  Order 458 

§  318.  Relations   between    the    Ray-Parameters   of   GAUSS   and 

SEIDEL 459 

§  322.  Conditions  of  the  Abolition  of  the  Spherical  Aberrations  of 

the  3rd  Order 467 

Art.  103.     Elimination  of  the  Magnitudes  Denoted  by  h,  u,  §  323. .  .   468-470 
Art.  104.     Remarks  on  SEIDEL'S  Formulae;  and  References  to  Other 

General  Methods,  §§  324-326 47o~473 


Contents.  xxi 

PAGE 

CHAPTER  XIII. 

Colour-Phenomena,  ARTS.  105-113,  §§  327~359 474~53* 

I.  DISPERSION  AND  PRISM-SPECTRA,  ARTS.  105-107, 

§§  327-342 474-503 

Art.  105.     Introductory  and  Historical,  §§  327-330 474-484 

§  327.  Relation  between  the  Refractive   Index  and  the  Wave- 

Length 474 

§  328.  NEWTON'S  Prism-Experiments  and  the  FRAUNHOFER  Lines 

of  the  Solar  Spectrum 475 

§  329.  The  Jena  Glass 478 

§  330.  Combinations  of  Thin  Prisms 483 

Art.  1 06.     The  Dispersion  of  a  System  of  Prisms,  §§  331-335 484-492 

§  332.  Dispersion  of  a  Single  Prism  in  Air 487 

§  333.  The  Dispersion  of  a  Train  of  Prisms  composed  alternately 

of  glass  and  air 488 

§  334.  Achromatic  Prism-Systems 489 

§  335.  Direct-Vision  Prism-System 491 

Art.  107.  Purity  of  the  Spectrum.  Resolving  Power  of  Prism-Sys- 
tem, §§  336-342 492-503 

§  336.  Measure  of  the  Purity  of  the  Spectrum 492 

§  337-  Purity  of  Spectrum  in  case  of  a  Single  Prism 494 

§  338.  Diffraction-Image  of  the  Slit 495 

§  339-  Ideal  Purity  of  the  Spectrum 497 

§  340.  Resolving  Power  of  Prism-System 498 

II.  THE  CHROMATIC  ABERRATIONS,  ARTS.  108-113, 

§§  343-359 503-531 

Art.  108.     The  Different  Kinds  of  Achromatism,  §  343 503-505 

Art.  109.  The  Chromatic  Variations  of  the  Position  and  Size  of  the 
Image,  in  Terms  of  the  Focal  Lengths  and  Focal  Distances 

of  the  Optical  System,  §§  344,  345 505-508 

Art.  no.  Formulae  Adapted  to  the  Numerical  Calculation  of  the 
Chromatic  Variations  of  the  Position  and  Magnifications 
of  the  Image  of  a  given  Object  in  a  Centered  System  of 

Spherical  Refracting  Surfaces,  §§  346-348 508-513 

§  346.  Chromatic  Longitudinal  Aberration 508 

§  347.  Differential  Formulae  for  the  Chromatic  Variations 510 

Art.  in.     Chromatic  Variations  in  Special  Cases,  §§  349-353 5I3~522 

§  349-  Optical  System  consisting  of  a  Single  Lens,  surrounded  on 

both  sides  by  air 513 

§  350.  Infinitely  Thin  Lens 516 

§  351 .  Chromatic  Aberration  of  a  System  of  Infinitely  Thin  Lenses  517 


xxii  Contents. 

PAGE 

§  352.  Two  Infinitely  Thin  Lenses  in  Contact 519 

§  353-  System  of   Two    Infinitely  Thin  Lenses    Separated   by  a 

Finite  Interval  (d) 520 

Art.  112.     The  Secondary  Spectrum,  §§  354,  355 523-526 

Art.  113.     Chromatic  Variations  of  the  Spherical  Aberrations,  §§  356- 

359 526-531 

CHAPTER  XIV. 

The  Aperture  and  the  Field  of  View.    Brightness  of  Optical 

Images,  ARTS.  114-123,  §§  360-396 532-582 

Art.  114.  The  Pupils,  §§  360-364.  . 532-540 

§  360.  Effect  of  Stops 532 

§  361.  The  Aperture-Stop 533 

§  363.  The  Aperture-Angle 538 

§  364.  The  Numerical  Aperture 538 

Art.  115.  The  Chief  Rays  and  the  Ray-Procedure,  §§  365-367 540-544 

§  365.  Chief  Ray  as  Representative  of  Bundle  of  Rays 540 

§  366.  Optical  Measuring  Instruments 541 

Art.  116.  Magnifying  Power,  §§  368,  369 544-549 

§  368.  The  Objective  Magnifying  Power 544 

§  369.  The  Subjective  Magnifying  Power 545 

Art.  117.  The  Field  of  View,  §§  370,  371 549~55i 

§  370.  Entrance- Port  and  Exit-Port 549 

Art.  1 1 8.     Projection-Systems  with  Infinitely  Narrow  Aperture  (®=o), 

§§  372-374 551-554 

§  372.  Focus-Plane  and  Screen-Plane 551 

§  373.  Perspective-Elongation 553 

§  374.  Correct  Distance  of  Viewing  a  Photograph 554 

Art.  119.     Optical  Systems  with  Finite  Aperture,  §§  375-381 555-5^3 

§  375'  Projected  Object  and  Projected  Image  in  the  case  of  Pro- 
jection-Systems of  Finite  Aperture 555 

§  377.  Focus-Depth  of  Projection-System  of  Finite  Aperture 557 

§  379.  Lack  of  Detail  in  the  Image  due  to  the  Focus-Depth 560 

§  380.  Focus-Depth  of  Optical  Systems  of  Finite  Aperture  used  in 

Conjunction  with  the  Eye 560 

§  381.  Accommodation-Depth 561 

Art.  1 20.    The  Field  of  View  in  the  case  of  Projection-Systems  of 

Finite  Aperture,  §§  382-386 563~57I 

§  382.  Case  of  a  Single  Entrance-Port 563 

§  383.  Case  of  Two  Entrance-Ports 568 


Contents.  xxiii 

PAGE 

INTENSITY  OF  ILLUMINATION  AND  BRIGHTNESS, 

ARTS.  121-123,  §§  387-396 571-582 

Art.  121.     Fundamental  Laws  of  Radiation,  §§  387-389 571-575 

§  387.  Radiation  of  Point-Source 571 

§  388.  Radiation  of  Luminous  Surface-Element 573 

§  389.  Equivalent  Light-Source rr.  574 

Art.  122.     Intensity  of  Radiation  of  Optical  Images,  §§  390-393 575~579 

§  390.  Optical  System  of  Infinitely  Narrow  Aperture  (Paraxial 

Rays) 575 

§  391.  Optical  System  of  Finite  Aperture 576 

§  393-  The  Illumination  in  the  Image-Space 578 

Art.  123.     Brightness  of  Optical  Images,  §§  394-396 579~582 

§  394.  Brightness  of  a  Luminous  Object 579 

§  396.  Brightness  of  a  Point-Source 581 

APPENDIX. 

Explanations  of  Letters,  Symbols,  Etc 583-612 

I.  DESIGNATIONS  OF  POINTS  IN  THE  DIAGRAMS 583-593 

II.   DESIGNATIONS  OF  LINES 593,  594 

III.  DESIGNATIONS  OF  SURFACES 594~596 

IV.  SYMBOLS  OF  LINEAR  MAGNITUDES 596-604 

V.   SYMBOLS  OF  ANGULAR  MAGNITUDES 604-608 

VI.  SYMBOLS    OF    NON-GEOMETRICAL    MAGNITUDES    (CONSTANTS, 

CO-EFFICIENTS,  FUNCTIONS,  ETC.) 608-612 


Index 613-626 


GEOMETRICAL  OPTICS. 


CHAPTER  I. 

METHODS  AND  FUNDAMENTAL  LAWS  OF  GEOMETRICAL  OPTICS. 
ART.  1.     THE  THEORIES   OF  LIGHT. 

I.  According  to  the  Corpuscular  or  Emission  Theory  of  Light, 
maintained  and  developed  by  NEWTON,  the  sensation  of  Light  is  due 
to  the  impact  on  the  retina  of  very  minute  particles,  or  corpuscles, 
projected  from  a  luminous  body  with  enormous  speeds  and  proceed- 
ing in  straight  lines.     Thus,  in  NEWTON'S  famous  work  on  Optics,1 
published  in  1704,  he  asks:  "Are  not  rays  of  light  very  small  bodies 
emitted  from  shining  substances?     For  such  bodies  will  pass  through 
uniform  mediums  in  right  lines  without  bending  into  the  shadow, 
which  is  the  nature  of  the  rays  of  light."     Opposed  to  this  view  was 
the  Undulatory  Theory  of  Light,  which,  notwithstanding  the  specu- 
lations that  have  been  found  in  the  writings  of  earlier  philosophers, 
such  as  LEONARD:  DA  VINCI,  GALILEO  and  others,  must  beyond  doubt 
be  attributed  to  HUYGENS  as  its  author,  whose  work,  entitled  Traite 
de  la  lumiere  (Ley den,  1690),  was  based  on  the  assumption  that  the 
phenomena  of  light  were  dependent  on  an  hypothetical  Ether,  or  very 
subtle,  imponderable  and  exceedingly  elastic  medium,  which  not  only 
pervaded  all  space  but  penetrated  freely  all  material  bodies  solid, 
liquid  and  gaseous.     According  to  this  theory,  the  something  that  was 
emitted  from  a  luminous  body  was  not  matter  at  all  but  a  kind  of 
Wave-Motion  which  was  propagated  through  the  all-pervading  ether 
with  a  finite  speed  which  is  different  according  to  the  different  cir- 
cumstances in  which  the  ether  through  which  the  disturbance  advanced 
is  conditioned  by  the  presence  of  ordinary  gross  matter.     This  remark- 
able and  ingenious  theory  encountered  at  first  great  difficulties,  and 
even  HUYGENS  himself  was  not  able  to  give  satisfactory  explanations 
of  some  of  the  most  familiar  phenomena  of  light.     In  the  end,  however, 
it  was  destined  to  triumph,  and,  in  the  hands  of  such  advocates 
as  YOUNG  and  above  all  of  FRESNEL  (who,  in  order  to  account  for  the 

I 1.  NEWTON:  Opticks:  or  a  treatise  of  the  reflexions,  inflexions,  and  colours  of  light  (Lon- 
don, 1704);  see  Book  iii.,  Qu.  29. 

2  1 


2  Geometrical  Optics,  Chapter  I.  [  §  4. 

Polarisation  of  Light,  was  led  to  assume  that  the  ether-vibrations  were 
transversal),  the  Wave-Theory  won  its  way  to  the  front  rank  of  science, 
where  it  remains  to-day  more  firmly  established  than  ever. 

The  Electromagnetic  Theory  of  Light,  which  is  a  development  from 
the  Wave-Theory,  is  a  monument  to  the  genius  and  mathematical 
insight  of  MAXWELL,  but  the  experimental  basis  of  this  theory  is  to 
be  found  in  the  investigations  of  HERTZ,  who  showed  that  electrical 
energy  also  was  propagated  by  means  of  ether- waves  which,  under 
certain  circumstances,  obeyed  the  laws  of  Reflexion  and  Refraction 
and  travelled  with  the  speed  of  light. 

2.  But,  independent  of  any  of  the  theories  as  to  the  real  nature 
of  light,  there  are  certain  well-ascertained  facts  about  the  mode  of 
propagation  of  light  which  may  themselves  be  made  the  basis  of  a 
certain  science  of  light,  and  which — provided  we  are  careful  to  con- 
fine our  investigations  along  these  lines  within  justifiable  limits — will 
lead  us  often  by  the  easiest  route  to  a  true  knowledge,  at  any  rate,  of 
the  behaviour  and  effects  of  light.     Moreover,  these  cardinal  facts, 
which  we  may  call  the  fundamental  characteristics  of  the  mode  of 
propagation  of  light,  are  so  few  and  so  simple  and  suffice  to  explain 
such  a  large  class  of  important  phenomena,  especially  those  phenomena 
on  which  the  design  and  construction  of  optical  instruments  chiefly 
depend,  that  the  advantage  of  this  method  has  been  long  recognized. 

ART.  2.      THE   SCOPE   AND   PLAN   OF  GEOMETRICAL   OPTICS. 

3.  The  fundamental  characteristics  of  the  mode  of  propagation  of 
light  may  be  enumerated  under  three  heads  as  follows: 

(i)  The  Law  of  the  Rectilinear  Propagation  of  Light,  from  which 
we  derive  the  ideas  of  "rays"  of  light;  (2)  the  assumption  that  the 
parts  of  a  beam  of  light  are  mutually  independent;  so  that,  for  ex- 
ample, the  effect  produced  at  any  point  is  to  be  attributed  only  to  the 
action  of  the  so-called  "rays"  which  pass  through  that  point;  and, 
finally,  (3)  The  Laws  of  Reflexion  and  Refraction  of  Light. 

These  laws,  inasmuch  as  they  are  concerned  essentially  only  with 
the  direction  of  the  propagation  of  light,  are  purely  geometrical;  and, 
hence,  the  science  which  is  based  upon  them,  and  which  seeks,  by  their 
means,  to  explain  the  phenomena  of  light,  either  as  they  occur  in  na- 
ture or  as  they  are  produced  by  the  agency  of  optical  instruments,  is 
called  Geometrical  Optics. 

4.  But  while  it  is  the  peculiar  office  of  Geometrical  Optics  to  give 
as  far  as  possible  explanations  of  such  phenomena  of  light  as  depend 
simply  on  changes  in  the  directions  in  which  the  light  is  propagated,  it 


§  5.]  Fundamental  Laws  of  Geometrical  Optics.  3 

does  not  pretend  to  be  able  to  explain  all  such  phenomena;  and, 
especially,  it  excludes  as  outside  of  its  province  all  cases  in  which  the 
light  is  propagated  in  anisotropic  or  crystalline  media. 

Moreover,  also,  although  the  fundamental  laws  above-mentioned  are 
sufficient  in  themselves  to  construct  a  very  complete  and  satisfactory 
system  of  explanation  of  a  large  class  of  optical  phenomena,  it  must 
not  be  supposed  that  Geometrical  Optics  is  willing  to  dispense  entirely 
or  even  partially  with  the  more  accurate  ideas  and  conceptions  which 
are  to  be  derived  only  by  the  consideration  of  the  real  and  essential 
nature  of  light.  If  such  were  to  be  our  procedure,  we  should  often  go 
astray,  and,  indeed,  we  know  by  experience  that  when  Geometrical 
Optics  has  ignored  or  even  lost  sight  of  the  notions  of  the  Wave-Theory 
of  Light,  and  pushed  too  far  the  geometrical  consequences  of  the  fun- 
damental theorems  on  which  it  is  based,  erroneous  results  have  been 
obtained.  On  the  contrary,  the  wave-phenomena  of  interference  and 
the  like  must  be  kept  throughout  constantly  in  view  even  when  they 
are  not  paraded  to  the  front,  and  every  result  should  be  subjected  to 
the  test  of  the  methods  of  Physical  Optics.  Viewed  in  this  way, 
Geometrical  Optics  is  not  to  be  regarded  as  a  mere  mathematical 
discipline — as  is  sometimes  said  by  way  of  reproach — but  it  takes  its 
rank  as  a  useful  and  important  branch  of  Physics. 

ART.  3.     THE   RECTILINEAR   PROPAGATION    OF   LIGHT. 

5.  In  an  isotropic  medium  light  travels  in  straight  lines,  is  the  state- 
ment of  a  fact,  which,  if  not  absolutely  and  unexceptionally  true, 
certainly  cannot  be  far  from  the  truth;  and,  indeed,  until  compara- 
tively recent  times  this  statement  had  never  been  called  in  question. 
The  fact  is  confidently  assumed  not  merely  in  the  ordinary  affairs  of 
life  but  in  the  most  exact  measurements  both  in  Geodesy  and  in  Astron- 
omy, and,  so  far  as  these  sciences  are  concerned,  its  validity  has  never 
been  doubted.  In  order  to  view  a  star  through  a  long  narrow  tube, 
the  axis  of  the  tube  must  be  pointed  so  that  it  coincides  with  the 
straight  line  which  joins  the  (real  or  apparent)  position  of  the  star 
with  the  eye  of  the  observer.  In  aiming  a  rifle  or  in  any  of  the  proc- 
esses that  we  call  "sighting"  the  method  is  based  with  certainty  upon 
this  commonest  fact  of  experience.  The  most  conclusive  proof  that 
a  line  is  straight  consists  in  showing  that  it  is  the  path  which  light 
pursues.  The  greatest  difficulty  that  HUYGENS  encountered  in  his 
wave-theory  of  light  was  to  explain  its  apparent  rectilinear  propaga- 
tion. It  was  from  this  law  that  the  idea  of  a  "ray  of  light"  originated. 

Nevertheless,  the  law  is  only  approximately  true,  as  has  been  well 


4  Geometrical  Optics,  Chapter  I.  [  §  6. 

ascertained  now  for  more  than  a  century.  For  when  we  proceed  to 
subject  it  to  as  rigid  a  test  as  possible,  and  try,  by  means  of  screens 
with  very  narrow  openings,  to  separate  from  a  beam  of  light  the  so- 
called  "rays"  themselves,  we  discover  that  these  latter  have  in  reality 
no  physical  existence;  and  that  the  narrower  we  succeed  in  making 
the  opening,  the  less  do  we  realize  the  idea  conveyed  by  the  term 
"ray".  When  the  light  arrives  at  the  narrow  opening,  it  does  not 
merely  pass  through  it  without  changing  its  direction,  but  it  spreads 
out  laterally  as  well,  utterly  misbehaving  itself  so  far  as  the  law  of 
rectilinear  propagation  is  concerned.  Thus,  although  the  straight 
line  joining  a  point-source  of  light  with  an  eye  may  pierce  an  inter- 
posed screen  at  an  opaque  part  of  the  screen,  a  narrow  slit  in  another 
part  of  the  screen  may  enable  the  eye  to  perceive  the  source.  When 
an  opaque  object  is  interposed  between  a  point-source  of  light  and  a 
screen,  the  shadow  on  the  screen  will  be  found  to  correspond  less  and 
less  with  the  geometrical  shadow  in  proportion  as  the  dimensions  of 
the  opaque  body  are  made  smaller  and  smaller,  and,  in  fact,  the  very 
places  where,  on  the  hypothesis  of  the  rectilinear  propagation  of  light, 
we  should  expect  shadows  often  prove  to  be  places  of  quite  contrary 
effects,  and  vice  versa.  The  fact  is,  light  is  propagated  not  by  "rays" 
but  by  waves,  and  the  rectilinear  propagation  of  light  is  practically  true 
in  general  because  the  wave-lengths  of  light  are  so  minute.  But  when 
we  have  to  do  with  narrow  apertures  and  obstacles  whose  dimensions 
are  comparable  with  those  of  the  wave-lengths,  we  have  the  so-called 
Diffraction-effects  which  are  treated  at  great  length  in  works  on  Physi- 
cal Optics  and  which  can  only  be  alluded  to  here. 

6.  However,  in  order  to  arrive  at  a  clear  comprehension  of  the 
matter,  let  us  consider  briefly  the  explanation  afforded  by  the  wave- 
theory  of  the  mode  of  propagation  of  light  in  an  isotropic  medium. 
We  may  begin  by  giving  Huygens's  Construction  of  the  Wave-Front, 
which  enables  us  to  see  how  HUYGENS  himself  tried  to  explain  the 
assumed  rectilinear  propagation  of  light. 

Let  0  (Fig.  i)  be  a  point-source1  of  light,  or  a  luminous  point,  from 
which  as  a  centre  or  origin  ether-waves  proceed  with  equal  speeds  in 
all  directions.  At  the  end  of  a  certain  time  the  disturbances  will  have 
arrived  at  all  the  points  which  lie  on  a  spherical  surface  a  described 
around  the  centre  O,  which  is  the  locus  of  all  the  points  in  the  iso- 

1  An  actual  "  point-source  "  of  light  by  itself  cannot  be  physically  realized.  A  "  lumi- 
nous point  "  is  an  infinitely  small  bit  of  luminous  surface.  Nevertheless,  exactly  as  in 
Mechanics  we  are  accustomed  to  speak  of  "  particles  of  matter",  and  similarly  in  all 
branches  of  Theoretical  Physics,  we  may  make  use  in  Optics  of  this  convenient  and  useful 
conception,  whether  it  be  actually  realizable  or  not. 


§6.] 


Fundamental  Laws  of  Geometrical  Optics. 


FIG.  1. 

HUYGENS'S  CONSTRUCTION  OF  THE  WAVE- 
FRONT. 


tropic  medium  that  are  in  this  particular  initial  phase  of  vibration, 

and  which  is  the  Wave- Front  at  this  instant.     According  to  HUYGENS, 

every  point  P  in  the  wave-front, 

from  the  instant  that  the  disturb- 
ance reaches  it,  will  become  a  new 

source  or    centre  of    disturbance, 

from  which  secondary  waves  will  be 

propagated  in  all  directions.    More- 
over, HUYGENS  assumed  that  these 

secondary  waves,  originating  at  all 

the  points  affected  by  the  principal 

wave,  interfere  with  each  other  in 

such   fashion   that   their  resultant 

sensible  effects  are  produced  only 

at  the  points  of  the  surface  which 

envelops  at  any  given  instant  all 

the  secondary  wave-fronts,  and  that 

this  enveloping  surface  is,  therefore, 

the    wave-front    at    that    instant. 

Obviously,  in  an  isotropic  medium,  such  as  is  here  supposed,  this  sur- 
face will  be  a  sphere  described  around  0 
as  centre. 

Accordingly,  if  waves  diverge  from  a 
luminous  point  0  (Fig.  2),  and  if  an 
opaque  plane  screen  HJ  with  an  opening 
A  B  is  interposed  in  front  of  the  advanc- 
ing waves,  the  wave-front  at  any  time  t 
may  be  constructed  as  follows :  Consider 
all  the  points,  such  as  M,  which  lie  in 
the  plane  of  the  screen  at  the  place 
where  the  aperture  is  made.  As  soon 
as  the  disturbance  arrives  at  one  of  these 
points,  it  will  become  a  new  centre  of 
disturbance,  from  which  will  diverge, 
therefore,  secondary  spherical  waves.  In 
general,  the  radii  of  these  secondary 
waves  will  be  different.  Thus,  in  the 
diagram,  as  here  drawn,  the  point  des- 
ignated by  A  is  nearer  the  source  O 
than  the  point  designated  by  M,  so  that 

the  disturbance  must  arrive  at  A   first,   and  hence  the  secondary 


HUYGENS'S  CONSTRUCTION  OF  THE 
WAVE-FRONT.  Spherical  waves  di- 
verging from  the  point-source  O  and 
passing  through  an  opening  AS  in 
the  opaque  screen  HJ.  The  arc  DC 
is  a  section  of  the  portion  of  the  spher- 
ical wave-front  <r  which  contains  the 
points  beyond  the  screen  which  have 
been  reached  by  the  disturbance. 


Geometrical  Optics,  Chapter  I. 


116. 


wave  proceeding  from  A  will  have  had  time  to  travel  farther  than 
the  secondary  wave  originating  at  M.  If  we  put  OM—  x,  and  if  we 
denote  the  radius  of  the  secondary  spherical  wave  around  M  at  the 
time  t  by  r,  then  d  =  x  +  r  will  denote  the  distance  from  0  to  which 
the  disturbance  is  propagated  in  the  time  /,  which  shows  that  as  x  in- 
creases, r  decreases;  that  is,  the  greater  is  the  distance  of  the  point  M 
from  the  source  O,  the  smaller  will  be  the  radius  of  the  secondary  wave- 
surface  around  this  point  M.  Thus,  the  enveloping  surface  is  seen  to 
be  the  portion  of  a  spherical  surface  of  radius  d  around  0  as  centre: 
it  is  that  part  of  this  spherical  surface  which  is  comprised  within  the 
cone  which  has  0  for  its  vertex  and  the  opening  A  B  of  the  screen  for 
its  base.  The  wave  proceeds,  therefore,  from  0  into  the  space  on  the 
other  side  of  the  screen,  but  on  this  side  of  the  screen  the  wave-sur- 
face is  limited  by  the  rays  drawn  from  O  to  the  points  in  the  edge  of 
the  opening.  According  to  HUYGENS'S  view,  the  disturbance  is  propa- 
gated within  this  cone  just  as  though  the  screen  were  not  interposed 

at  all,  whereas  points  on  the  far  side  of 
the  screen  but  outside  this  cone  of  rays 
are  not  affected  at  all.  This  mode  of  ex- 
planation leads  to  the  theory  of  the  recti- 
linear propagation  of  light. 

If  the  luminous  point  O  (Fig.  3)  is  at  such 
a  distance  from  the  screen  that  the  dimen- 
sions of  the  opening  A  B  may  be  regarded 
as  vanishingly  small  in  comparison  there- 
with, we  shall  have  a  cylindrical  bundle  of 
rays,  and  the  wave-fronts  will  be  plane  in- 
stead of  spherical.1 

The    most   obvious   objection   to   HUY- 
GENS'S construction  is,  What  right  has  he 
to  assume  that  the  points  of  sensible  effects 
are  the  points  on  the  surface  which  envelops 
the  secondary  waves?     And  why  is  the  light  not  propagated  backwards 

1  The  single  points  of  a  luminous  heavenly  body  are  to  be  regarded  as  at  an  infinite 
distance  in  comparison  with  the  dimensions  of  our  apparatus,  so  that  the  wave-fronts  of 
the  disturbances  emitted  from  such  points  are  plane.  But  the  rays  which  come  from  dif- 
ferent points  of  a  celestial  body  cannot  be  regarded  as  parallel  unless  the  parallax  of  the 
star  is  sufficiently  small.  This  angle  has  a  right  considerable  magnitude  in  the  cases  of 
both  the  sun  and  the  moon,  so  that  the  divergence  of  the  rays  which  come  from  opposite 
ends  of  the  diameters  of  these  bodies  may  amount  to  more  than  half  a  degree.  For  most 
experiments  in  Optics  this  divergence  is  negligible,  and  a  beam  of  sunlight  may  be  regarded 
as  consisting  of  parallel  rays.  We  may  obtain  bundles  of  parallel  rays  from  terrestrial 
sources  of  light  by  means  of  lenses,  etc. 


FIG.  3. 

HUYGENS'S  CONSTRUCTION  OF 
THE  WAVE- FRONT.  Plane  waves 
proceeding  through  an  opening  in 
a  screen. 


§  7.]  Fundamental  Laws  of  Geometrical  Optics.  7 

as  well  as  forwards?  Moreover,  if  the  opening  in  the  screen  is  very 
narrow,  this  construction  does  not  correspond  at  all  with  the  observed 
facts. 

7.  FresnePs  Extension  of  Huygens's  Method.  In  place  of  HUY- 
GENS'S  arbitrary  assumption  that  the  places 
where  there  are  sensible  effects  are  to  be  found 
only  on  the  surface  which  envelops  the  ele- 
mentary waves,  FRESNEL  insisted  that  these 
secondary  waves,  encountering  each  other, 
must  therefore  be  regarded  as  interfering  with 
each  other,  and  thus  he  conceived  that  the 

•n  /T-"          \  i-  FRESNEL'S  METHOD.     The 

disturbance  at  any  point  P  (r  ig.  4)  must  be  effect  at  the  point  p  of  a  dis- 
due  to  the  superposition  of  the  component  turbance  originating  at  o  is  to 

_    ,  be  attributed  almost  entirely 

disturbances  propagated  to  P  from  all  the  to  the  disturbance  that  is  prop- 
points  Of  the  Wave-SUrface  <7.  According  tO  agated  along  the  straight  line 
L,  °  ,  OP;  provided  the  wave-lengths 

FRESNEL,   therefore,   light-effects  are   to    be     are  very  small, 
found,  not  on  the  enveloping  surface,  but  at 

all  points  where  the  secondary  waves  combine  to  reinforce  each  other. 
On  investigation — which  we  do  not  attempt  to  show  here — it  ap- 
pears that  the  disturbances  which  arrive  at  P  along  all  the  straight 
lines  joining  P  with  points  on  the  wave-front  in  great  measure  neu- 
tralize each  other,  and  the  result  is  (assuming  that  the  wave-length 
is  small)  that  the  actual  effect  at  P  may  be  considered  as  due  wholly  to  the 
action  of  a  very  small  element  of  the  wave-front  situated  at  the  point  A 
where  the  straight  line  joining  0  with  P  intersects  the  wave-front  a. 
(This  point  A  is  called  the  "pole"  of  the  wave  with  respect  to  the 
point  P ;  it  is  the  point  of  the  wave-front  that  is  nearest  to  P,  so  that 
the  disturbance  from  this  point  arrives  at  P  before  the  disturbance 
from  any  other  point  of  the  wave-front.)  Hence,  if  between  0  and  P 
we  interpose  a  small  opaque  screen  which  exactly  shuts  off  from  P 
the  effect  due  to  the  small  "zone"  around  A,  there  will  be  darkness  at 
P;  moreover,  what  is  true  of  this  point  P  is  true  also  of  any  point 
which,  like  P,  is  situated  on  the  straight  line  OP.  On  the  other  hand 
if  a  plane  screen  is  placed  tangent  to  the  wave-surface  at  A,  with  a 
small  circular  opening  in  it  the  centre  of  which  is  at  A,  so  that  the 
point  P  is  screened  from  the  entire  wave-surface  except  the  very  small 
"effective  zone"  immediately  around  A,  the  effect  at  P,  as  also  at  all 
points  along  the  straight  line  AP,  is  found  to  be  precisely  the  same 
as  though  the  screen  had  not  been  interposed.  It  is  thus  that  the 
idea  of  HUYGENS  as  developed  by  FRESNEL  leads,  as  we  see,  to  the 
theory  of  the  approximate  rectilinear  propagation  of  light — that  is, 


8  Geometrical  Optics,  Chapter  I.  [  §  9. 

light  does   in   fact   behave  very  nearly  as  if  it  were  propagated  in 
straight  lines. 

8.  Although,  therefore,  this  fundamental  law  has  always  to  be 
stated  with  certain  reservations,  and,  as  a  matter  of   fact,  is  never 
strictly  true,  yet  even  when  it  is  regarded  from  the  standpoint  of  the 
wave-theory*  the  law  of  the  rectilinear  propagation  of  light  loses  very 
little  of  its  meaning.     On  the  contrary,  in  agreement  with  experience, 
that  theory  shows  that  in  the  cases  which  ordinarily  occur,  especially 
in  those  cases  where  we  have  to  do  with  beams  of  light  of  finite  di- 
mensions, the  effects  at  any  rate  are  for  all  practical  purposes  the 
same  as  if  these  beams  of  light  were  composed  of  separate  rays,  each 
independent  of  the  others,  along  which  the  light  is  propagated  in 
straight  lines.     But,  however  useful  and  generally  safe  this  simple  and 
convenient  rule  may  be,  it  must  be  borne  in  mind  that  it  is  inexact 
and  we  must  be  prepared,  therefore,  to  meet  here  and  there  excep- 
tional cases  where  the  rule  is  plainly  inadmissible.     It  is  only  in  this 
way  that  the  methods  of  Geometrical  Optics  can  be  approved. 

ART.  4.     RAYS  OF  LIGHT. 

9.  A  self-luminous  point  is  said  to  emit  "rays  of  light"  in  all  direc- 
tions.    In  an  isotropic  medium  (§10)  the  ray-paths  are  straight  lines 
proceeding  from  the  centre  of  the  expanding  spherical  wave-surface; 
and  whether  the  medium  is  isotropic  or  not,  the  direction  of  the  ray- 
path  at  any  point  is  to  be  considered  as  being  always  along  the  normal 
to  the  wave-surface  that  goes  through  that  point  (see  §42).     What 
are  called  "rays  of  light"  in  Geometrical  Optics  are  in  fact  those  short- 
est paths,  optically  speaking  (§38),  along  which  the  ether-disturbances 
are  propagated.     Employed  in  this  sense,  the  word  "ray"  is  a  purely 
geometrical  idea.     However,  there  is  a  certain  sense  in  which  we  can 
attach  a  physical  meaning  also  to  these  so-called  "light-rays".     For, 
as  a  rule,  it  is  approximately  true  that  the  ether-disturbance  at  any 
point  of  the  path  of  a  ray  of  light  is  due  to  disturbances  which  have 
occurred  successively  at  all  points  along  the  ray  that  are  nearer  to  the 
source  than  the  point  in  question;  so  that,  according  to  this  view, 
the  effect  at  any  point  P  is  to  be  considered  as  in  no  degree  arising 
from  disturbances  at  other  points  which  do  not  lie  on  a  ray  passing 
through  P.  This  is,  in  fact,  the  Principle  of  the  Mutual  Independence 
of  the  Rays  of  Light,  which  is  also  one  of  the  fundamental  laws  of 
Geometrical  Optics,  and  which  assumes  that  each  ray  in  a  beam  of 
light  is  somehow  separate  and  distinct  from  its  fellows,  and  has,  there- 
fore, a  certain  physical  existence.     Thus,  for  example,  if  we  have  a 


§  10.]  Fundamental  Laws  of  Geometrical  Optics.  9 

wide-angle  cone  of  rays  incident  on  a  screen  and  producing  there  a 
comparatively  large  light-spot,  and  if  we  interpose  an  opaque  object 
so  as  to  intercept  a  considerable  fraction  of  the  rays  before  they 
reach  the  screen,  a  corresponding  portion  of  the  light  on  the  screen 
will  vanish;  and,  hence,  it  can  be  inferred  that  we  may  suppress  some 
of  the  rays  in  a  beam  without  altering,  apparently,  the  effect  pro- 
duced by  the  remaining  rays. 

Here  also,  however,  when  this  principle  is  examined  from  the  stand- 
point of  the  wave-theory,  we  find  that  it,  too,  has  to  be  stated  with 
reservations.  According  to  FRESNEL,  the  disturbance  at  the  point  P 
(Fig.  4)  is  to  be  considered  as  the  resultant  of  an  infinite  number  of 
partial  disturbances  propagated  to  P  from  all  points  situated  on  the 
wave-front  a ;  so  that  in  a  certain  sense  P  may  be  considered  as  being 
at  the  vertex,  or  ''storm-centre",  of  a  cone  of  rays  which  are  by  no 
means  independent  of  each  other.  Every  point,  such  as  P,  which  lies 
ahead  of  the  advancing  wave-front  is  in  similar  circumstances.  But, 
as  has  been  stated  (§7),  the  resultant  effect  at  the  point  P  is  due  in 
the  main  to  the  disturbance  that  is  propagated  along  the  central  ray 
of  the  cone  of  rays  that  converge  to  P;  and,  thus,  the  law  of  the 
Mutual  Independence  of  Rays,  if  it  is  true  at  all,  can  only  be  said  to 
be  true  of  these  central  rays  of  all  such  cones  of  rays  as  are  here  meant. 
In  point  of  fact,  the  resultant  effect  at  the  point  P  is  to  be  ascribed 
not  merely  to  the  disturbance  propagated  along  this  central  ray  from 
the  pole  A  of  the  wave,  but  to  a  zone  of  the  wave-surface  of  very 
small,  but  finite,  dimensions,  with  its  vertex  at  A.  And  the  moment 
we  attempt  to  isolate  physically  the  ray  A  P  by  screening  P  from  the 
effects  of  this  zone,  the  effect  at  P  vanishes  entirely  and  the  ray  ceases 
to  exist. 

10.  It  is  best,  therefore,  without  any  reference  to  its  physical  mean- 
ing, to  define  a  ray  of  light  as  a  line  or  path  along  which  the  ether- 
disturbance  is  propagated.  An  optical  medium  is  any  space,  whether 
filled  or  not  with  ponderable  matter,  which  may  be  traversed  by  rays 
of  light.  In  Geometrical  Optics,  where  we  have  to  do  only  with  iso- 
tropic  media,  the  rays  of  light  are  straight  lines  (Art.  3).  At  a  sur- 
face of  separation  of  two  media  the  direction  of  the  ray  will  usually 
be  changed  abruptly,  either  when  the  ray  passes  from  one  medium  into 
the  next  or  is  bent  away  at  the  surface  of  a  body;  so  that  under  such 
circumstances  the  ray-path  will  consist  of  a  series  of  straight  line- 
segments.  If,  for  example,  Bk  designates  the  point  where  the  ray 
meets  the  &th  surface,  then  the  straight  line-segment  Bk_lBk  will 
represent  the  path  of  the  ray  in  the  kth  medium :  and  here  it  may  be 


10  Geometrical  Optics,  Chapter  I.  [  §  12. 

remarked  that,  so  long  as  we  are  speaking  of  this  portion  of  the  ray- 
path,  any  point  P  lying  on  the  straight  line  determined  by  Bk_^  and 
Bk  is  to  be  considered  as  situated  in  the  kth  medium,  even  though 
the  substance  of  which  the  medium  is  composed  does  not  extend  out 
to  the  point  P.  If  the  point  P  is  situated  on  the  straight  line  Bk_lBb 
between  these  two  incidence-points,  we  say  that  the  ray  in  this  medium 
passes  really  through  the  point  P\  otherwise,  we  say  that  the  ray 
goes  virtually  through  the  point  P. 

ART.  5.     THE  BEHAVIOUR  OF  LIGHT  AT  THE  SURFACE  OF  SEPARATION 
OF  TWO  ISOTROPIC  MEDIA. 

11.  In  order  to  have  clear  ideas  of  certain  matters  mentioned  in 
the  preceding  articles,  it  will  be  necessary  to  know  how  the  rays  of 
light  are  affected  when  they  arrive  at  the  boundary-surface  separating 
two  adjoining  optical  media.     At  such  a  surface  the  "incident"  light 
(as  it  is  called)  will,  in  general,  be  divided  into  two  portions,  which  are 
propagated  from  the   places  where  the  light  falls  on  the  surface  in 
abruptly  changed  directions: 

(1)  One  portion  of  the  light  is  turned  back  or  "reflected"  at  the  sur- 
face, and  pursues  its  progress  in  the  first  medium  along  new  ray-paths 
(except  under  special  conditions). 

(2)  The  remaining  portion,  crossing  the  surface  and  entering  the 
second  medium,  makes  its  way,  in  general,  in  this  new  region;  this 
is  the  so-called  "refracted"  light. 

12.  However,  here  also  a  closer  study  of  these  phenomena  reveals 
the  fact  that  neither  of  the  above  statements  is  an  entirely  accurate 
description.     Thus,  it  will  be  found  that  even  that  part  of  the  light 
which  is  said  to  be  reflected  and  which  ultimately  returns  into  the 
first  medium  had  crossed  the  boundary-surface  and  penetrated  a  little 
way  into  the  second  medium.     This  is  the  explanation  of  the  colour  of 
a  body  as  seen  by  reflected  light :  the  incident  light  falling  on  the  body 
and  penetrating  to  a  slight  extent  below  the  surface  is  there,  according 
to  the  "Theory  of  Selective  Absorption"  (into  which  we  cannot  enter 
here),  robbed  of  certain  of  its  constituent  parts,  and  only  the  remain- 
der is  finally  reflected.     The  depth  of  penetration  depends  on  the 
qualities  of  the  two  media  and  in  a  very  great  degree  on  the  character 
of  the  separating  surface.     Thus,  for  example,  if  the  second  medium 
is  glass,  this  question  will  involve  the  knowledge  of  whether  the  glass 
is  in  a  compact  (solid)  state  or  in  the  form  of  a  fine  powder ;  and  if  the 
glass  were  solid,  the  next  question  would  be  as  to  the  surface,  whether 
it  was  highly  polished  or  not,  etc. 


§  13.]  Fundamental  Laws  of  Geometrical  Optics.  11 

When  a  beam  of  sunlight  is  admitted  through  an  opening  in  a  shut- 
ter into  an  otherwise  dark  room,  and  is  allowed  to  fall,  for  example, 
on  a  metallic  surface,  the  reflected  light  itself  consists  of  two  portions, 
viz.,  one  part  (in  this  case  the  greater  part)  which  leaves  the  metallic 
surface  in  a  perfectly  definite  direction,  and  which  is  said  therefore 
to  be  regularly  reflected,  and  another  part  which  leaves  the  surface  in 
countless  different  directions,  and  which  is  said  to  be  "scattered". 
This  scattering  or  "diffusion"  of  the  reflected  light  is  due  to  the  in- 
equalities or  rugosities  of  the  surface;  it  may  be  greatly  diminished 
by  cleaning  and  polishing  the  surface.  If  the  reflecting  surface  is  geo- 
metrically regular  and  physically  smooth,  the  reflected  light  will  be 
nearly  all  regularly  reflected.  And  even  in  those  cases  where  the  light 
is  irregularly  reflected  or  diffused,  as,  for  example,  when  a  beam  of 
sunlight  is  reflected  from  a  ground-glass  surface,  it  would  be  more 
correct  to  attribute  the  irregularity  not  so  much  to  the  behaviour  of 
the  rays  of  light  as  to  the  peculiarity  of  the  surface  itself.  Perhaps, 
if  we  knew  precisely  the  arrangement  and  orientation  of  the  elements 
of  such  a  reflecting  surface,  we  should  discover  that  the  reflexion  was 
quite  regular  after  all.  However,  the  actual  dimensions  of  these 
rugosities  of  the  surface  will  also  affect  the  phenomenon,  inasmuch  as 
when  these  dimensions  are  sufficiently  small,  the  assumptions  which 
lie  at  the  foundation  of  Geometrical  Optics  will  cease  to  be  valid. 

It  is  in  consequence  of  this  fact,  that  the  light  which  is  incident  on 
a  rough  surface  is  subjected  to  different  experiences  at  the  different 
places  in  the  surface,  that  these  irregularities  are  made  visible  to  us 
as  themselves  sources  of  rays  of  light;  whereas  if  the  reflecting  sur- 
face were  perfectly  smooth,  so  that  the  rays  were  regularly  reflected 
all  according  to  the  same  law,  we  should  not  be  able  to  see  the  surface 
at  all,  we  should  see  merely  the  images  of  objects  from  which  the  rays 
had  come — objects  which  were  either  self-luminous  or  else  illuminated 
by  diffusely  reflected  light.  Moreover,  in  order  to  view  the  images, 
the  eye  would  have  to  be  placed  somewhere  along  these  special  routes 
of  the  reflected  rays;  otherwise,  none  of  these  rays  would  enter  the 
eye  and  nothing  would  be  visible  by  the  reflected  light.  Most  objects 
are  seen  by  diffusely  reflected  light,  and  no  matter  where  the  eye  is 
situated,  it  will  intercept  some  of  the  rays  that  are  scattered  from  the 
surface  of  the  body. 

13.  In  large  measure  the  above  observations  concerning  the  por- 
tion of  the  light  that  is  reflected  apply  also  to  the  other  portion  that 
is  refracted.  If  the  surface  of  separation  of  the  two  media  is  smooth, 
the  directions  of  the  refracted  rays  will,  in  general,  depend  only  on  the 


12  Geometrical  Optics,  Chapter  I.  [  §  13. 

directions  of  the  incident  rays  according  to  the  so-called  Law  of  Re- 
fraction; and  in  this  case  the  light  is  said  to  be  regularly  refracted. 
But  if  the  boundary-surface  is  rough,  the  rays  will  be  diffusely  re- 
fracted in  all  directions  ("irregular  refraction"). 

The  light  which  enters  the  second  medium  may  be  modified  in  vari- 
ous ways.  A  greater  or  less  portion  of  it,  depending  on  the  character 
and  peculiarity  of  the  medium,  will  be  absorbed;  that  is,  the  ether 
loses  some  of  its  energy  and  ordinary  matter  gains  it.  Invariably,  a 
fraction  of  the  light-energy  will  be  transformed  into  heat,  possibly 
also  into  chemical  and  electrical  forms  of  energy.  If  the  medium  is 
perfectly  transparent,  the  rays  of  light  traverse  it  without  being  ab- 
sorbed at  all;  whereas  if  the  medium  absorbs  all  the  light-rays,  it  is 
said  to  be  perfectly  opaque.  No  medium  is  absolutely  transparent 
on  the  one  hand  or  absolutely  opaque  on  the  other.  A  perfectly  trans- 
parent body  would  be  quite  invisible,  although  we  may  easily  be  made 
aware  of  the  presence  of  such  a  body  by  the  distortion  of  the  images 
of  bodies  viewed  through  it.  As  a  rule,  the  absorptive  power  of  a 
medium  will  depend  on  the  colour  (or  wave-length)  of  the  light.  Thus, 
a  piece  of  green  glass  will  allow  only  certain  kinds  of  light  to  pass 
through  it,  and  therefore  when  the  rays  of  the  sun  fall  on  it,  it  will 
absorb  some  of  these  rays  and  be  transparent  to  others,  and  the  trans- 
mitted light  falling  on  the  retina  of  the  eye,  will  produce  a  sensation 
which  we  describe  vaguely  as  green  light.  An  interesting  phenomenon 
occurs  called  Fluorescence,  whereby  the  colour  of  the  light  undergoes 
a  change  in  the  second  medium. 

Again,  there  are  some  media  which,  although  they  cannot  be  called 
transparent,  nevertheless  permit  light  to  pass  through  them  in  a  more 
or  less  irregular  and  imperfect  fashion;  for  example,  such  substances 
as  porcelain,  milk,  blood,  moist  atmospheric  air,  which  contain  sus- 
pended or  imbedded  in  them  particles  of  matter  of  a  different  optical 
quality  from  that  of  the  surrounding  mass.  The  light  undergoes  in- 
ternal diffused  reflexion  at  these  particles.  Objects  viewed  through 
such  media  can  be  discerned,  perhaps,  but  always  more  or  less  indis- 
tinctly. These  so-called  ' 'cloudy  media"  are  said,  therefore,  to  be 
translucent,  but  not  transparent. 

I  Ms  usually  assumed  in  Geometrical  Optics  that  the  media  are  not 
only  homogeneous,  but  perfectly  transparent;  and  also  that  the  sur- 
faces of  separation  between  pairs  of  adjoining  media  are  perfectly 
smooth. 


IS.] 


Fundamental  Laws  of  Geometrical  Optics. 


13 


ART.  6.     THE  LAWS  OF  REFLEXION  AND  REFRACTION. 

14.  Let   MM    (Fig.  5)  be  the  trace  in  the  plane  of  the  diagram  of 
the  smooth  reflecting  or  refracting  surface  separating  two  transparent 
iso tropic  media.     Let  PB  represent  the  rectilinear  path  of  a  ray  of 
light  in  the  first  medium  (a).     The  ray  PB  is  called  the  incident  ray, 
the  point  B  where  this  ray  meets  the  boundary-surface  between  the 
two  media  (a)  and  (b)  is  called  the  incidence-point,  the  normal  N  Nf 
to  the  surface  at  the  point  B  is  called  the  incidence-normal,  and  the 
plane  PB  N  determined  by  the  incident  ray  and  the  incidence-normal 
(which  is  here  the  plane  of  the  paper)  is  called  the  plane  of  incidence. 

In  general,  to  an  incident  ray  PB  there  will  correspond  two  rays, 
viz.,  a  reflected  ray  BR,  which  re- 
mains in  the  first  medium  (a)  and 
a  refracted  ray  BQ,  which  shows 
the  path  taken  by  the  light  in  the 
second  medium  (b).  The  acute 
angles  at  the  incidence-point  B  be- 
tween the  incidence  normal  NB  N' 
and  the  rays  PB,  BR  and  BQ 
are  called  the  angles  of  incidence,  re- 
flexion and  refraction,  respectively. 
Each  of  these  angles  is  defined  as 
the  acute  angle  through  which  the 
incidence-normal  has  to  be  turned  in 
order  to  bring  it  into  coincidence  with 
the  straight  line  which  shows  the  path 
of  the  ray  in  question.  Thus,  in  the 
diagram  the  angles  of  incidence,  re- 
flexion and  refraction  are  ^  NBP, 
Z  NBRand  /  N'B Q, respectively; 
where  the  order  in  which  the  let- 
ters are  written  indicates  the  sense 

of  rotation.     These  angles  are  to  be  reckoned  as  positive  or  negative 
according  as  the  sense  of  rotation  is  counter-clockwise  or  clockwise. 

15.  The  Laws  of  Reflexion  and  Refraction,  as  determined  by  ex- 
periment, may  now  be  set  forth  in  the  following  statements: 

(1)  Both  the  reflected  and  the  refracted  rays  lie  in  the  plane  of  inci- 
dence. 

(2)  The  reflected  ray  in  the  first  medium  and  the  refracted  ray  in  the 
second  medium  lie  on  the  opposite  side  of  the  normal  from  the  incident 
ray  in  the  first  medium.     Or  if  we  prolong  the  refracted  ray  backwards 


FIG.  5. 

I,AWS  OF  REFLEXION  AND  REFRACTION. 
MM-  is  a  section  in  the  plane  of  incidence  (plane 
of  paper)  of  the  surface  separating  the  first 
medium  (a)  from  the  second  medium  (b). 
The  point  B  is  the  point  of  incidence,  and 
NN'  is  the  normal  to  the  surface  at  this  point. 
PB,  BR  and  BQ  are  the  incident,  reflected  and 
refracted  rays,  respectively. 

Z  NBP=  o,     L  NBR  = — a.     Z  N'BQ  =  a'. 


14  Geometrical  Optics,  Chapter  I.  [  §  17. 

into  the  first  medium,  we  see  that  in  this  medium  the  straight  lines 
belonging  to  the  incident  and  refracted  rays  lie  on  the  same  side  of 
the  incidence-normal,  whereas  the  incident  and  reflected  rays  lie  on 
opposite  sides  of  the  normal.  Thus,  while  the  angles  of  incidence  and 
refraction  have  always  like  signs,  the  angles  of  incidence  and  reflexion 
have  opposite  signs. 

(3)  The  magnitudes  of  the  angles  of  incidence  and  reflexion  are  equal; 
that  is, 

Z  NBP  =  a  =  -  Z  NBR  =  Z  RB  N. 

(4)  The  sines  of  the  angles  of  incidence  and  refraction  are  in  a  con- 
stant ratio,  the  value  of  which  depends  only  on  the  nature  of  the  two  media 
which  are  separated  by  the  refracting  surface  and  on  the  wave-length  of 
the  light. 

Thus,  if  the  angles  of  incidence  and  refraction  are  denoted  by  a,  a', 
so  that  Z  NBP  =  a,  Z  N'BQ  =  a',  the  law  of  refraction  may  be 
expressed  by  the  following  formula: 


sn  a 


where  the  constant  ratio,  denoted  here  by  nab,  which  for  light  of  a 
given  wave-length,  as  has  been  stated,  depends  only  on  the  nature  of 
the  two  media  designated  by  the  letters  a  and  b,  is  called  the  relative 
index  of  refraction  from  the  medium  (a)  into  the  medium  (b)  ,  or  the  index 
of  refraction  of  medium  (b)  with  respect  to  medium  (a).  The  order 
in  which  the  subscripts  are  written  is  the  same  as  the  order  in  which 
the  media  are  traversed  by  the  light. 

16.  The  best  experimental  proof  of  the  law  of  reflexion  is  obtained 
by  the  use  of  a  theodolite  or  meridian  circle  to  observe  the  light  re- 
flected from  an  artificial  mercury-horizon.     This  is  the  actual  method 
employed  in  the  astronomical  measurement  of  the  altitude  of  a  star, 
and  is  capable  of  a  very  high  degree  of  accuracy. 

The  law  of  refraction  may  be  regarded  as  completely  verified  by  the 
methods  which  are  employed  in  the  determinations  of  the  values  of 
the  indices  of  refraction  for  different  pairs  of  media,  and,  above  all, 
in  the  design  and  construction  of  optical  instruments,  by  the  complete 
agreement  between  the  actual  performances  of  such  apparatus  and 
the  calculations  based  on  the  law  of  refraction. 

17.  The  law  of  the  reflexion  of  light  is  very  ancient.     The  earliest 
precise  statement  of  the  law  is  to  be  found  in  a  work  on  optics  attrib- 


§  18.]  Fundamental  Laws  of  Geometrical  Optics.  15 

uted  to  EUCLID  (300  B.  C.).  On  the  other  hand,  the  law  of  refraction 
is  much  more  modern.  CLAUDIUS  PTOLEM^EUS,  the  great  astrono- 
mer, who  flourished  during  the  reigns  of  the  ANTONINES,  published  a 
treatise  on  optics  ('OTTTUC^  Trpay/xareia)  in  which  he  describes  a  number 
of  experiments  whereby  he  measured  the  angles  of  incidence  and  re- 
fraction, without,  however,  discovering  the  law.  The  next  experi- 
ments along  this  line  of  which  we  have  any  record  are  those  of  ALHAZEN 
who  died  in  Cairo  in  1038  ;  he  repeated  the  experiments  of  PTOLEM^EUS, 
but  added  nothing  to  the  previous  knowledge  of  the  matter.  KEPLER 
also  made  experiments,  but  was  equally  unsuccessful.  The  real  dis- 
coverer of  the  law  was  WILLEBRORD  SNELL,  of  Leyden,  who  announced 
it  some  time  prior  to  1626.  It  was  first  published  by  DESCARTES1 
in  1637;  who  seems  undoubtedly  to  have  obtained  it  from  SNELL, 
although  he  failed  to  mention  his  name  in  connection  with  it. 

18.  In  the  case  of  Reflexion,  it  is  obvious  that  the  directions  of 
the  incident  and  reflected  rays  may  be  reversed,  so  that  if  PBR  (Fig.  5) 
represents  the  path  pursued  by  the  light  in  going  from  P  to  R,  under- 
going reflexion  at  the  incidence-point  B,  then  RBP  will  represent  the 
path  which  the  light  takes  in  going  from  R  to  P  under  the  same  cir- 
cumstances, that  is,  via  the  incidence-point  B.  Experiment  shows 
that  the  same  rule  holds  good  also  for  the  ray  refracted  at  B  ;  so  that 
if  PBQ  is  the  route  followed  by  a  ray  in  going  from  a  point  P  in  the 
medium  (a)  to  a  point  Q  in  the  medium  (b),  undergoing  refraction  at 
the  incidence-point  B,  the  same  route  will  be  pursued  in  the  reverse 
sense  QBP  by  a  ray  whose  direction  in  the  medium  (b)  is  from  Q 
towards  B.  And,  hence,  since 

sin  a  sin  a' 


we  have  obviously,  the  relation: 

nab-nba=i.  (2) 

This  general  law  of  optics,  known  as  the  Principle  of  the  Reversi- 
bility of  the  Light-Path,  may  be  stated  as  follows  : 

If  a  ray  of  light,  undergoing  any  number  of  reflexions  and  refrac- 
tions, pursues  a  certain  route  from  one  point  A  to  another  point  A', 
and  if  at  A'  it  is  incident  normally  on  a  mirror  so  that  it  is  reflected 

1  RENE  Du  PERRON  DESCARTES:  Discours  de  la  methode  pour  bien  conduire  sa  raison 
etchercherlaveritedans  Us  sciences;  plus  la  Dioptrique,  les  Meteor  eset  la  Geometric  (Leyden, 
1637). 


16 


Geometrical  Optics,  Chapter  I. 


[§19. 


back  from  A'  in  the  direction  exactly  opposite  to  that  by  which  it 
arrived,  it  will  return  over  precisely  the  same  route  in  the  reverse 
order  and  arrive  finally  at  A  again. 

19.  The  Laws  of  Reflexion  and  Refraction  as  derived  by  the 
Wave-Theory  (HUYGENS'S  Construction) .  A  plane  wave  travelling  in 
an  isotropic  medium  advances  with  uniform  speed  and  without  change 


FIG.  6. 

HUYGENS'S  CONSTRUCTION  OF  THE  REFLECTED  AND  REFRACTED  WAVE-FRONTS  IN  THE  CASE 
OF  A  PLANE  WAVE  INCIDENT  ON  A  PLANE  SURFACE.  MM  is  a  section  in  the  plane  of  incidence 
(plane  of  paper)  of  a  plane  surface  separating  the  first  medium  (a)  from  the  second  medium  (£). 
The  straight  lines  AB,  CG  and  C/fare  the  traces  in  the  plane  of  the  paper  of  the  incident,  reflected 
and  refracted  wave-fronts. 

of  form  along  the  direction  of  the  normal  to  its  plane,  so  that  the 
rays  are  parallel  to  each  other  and  perpendicular  to  the  plane  wave- 
front.  If  the  wave-front  arrives  at  a  smooth  geometric  surface  sepa- 
rating the  first  medium  (a)  from  another  isotropic  optical  medium  (6), 
the  refracted  wave  proceeding  into  this  second  medium  will,  in  general, 
be  changed  both  in  form  and  in  direction.  At  the  same  time  also  a 


§  19.]  Fundamental  Laws  of  Geometrical  Optics.  17 

wave  will  be  reflected  back  from  the  boundary-surface  into  the  first 
medium,  which  likewise  may  be  changed  both  in  form  and  in  direc- 
tion. But  the  speed  of  propagation  of  the  reflected  wave  will  be  the 
same  as  that  of  the  incident  wave ;  whereas  the  speed  of  propagation 
of  the  refracted  wave  in  the  new  medium  (6)  will  be  different  from 
that  of  the  incident  wave  in  the  medium  (a). 

For  the  sake  of  simplicity,  let  us  suppose  that  the  two  media  (a) 
and  (b)  are  separated  by  a  plane  surface.  We  proceed  to  give  HUY- 
GENS'S  Construction  of  the  Reflected  and  Refracted  Wave-Fronts  for 
this  case.  In  the  diagram  (Fig.  6)  /z/z  represents  the  trace  in  the  plane 
of  the  paper  of  the  plane  surface  separating  the  media  (a)  and  (6); 
and  A  B  is  the  trace  in  the  same  plane  of  a  portion  of  the  advancing 
incident  plane  wave;  so  that  the  incident  rays  in  the  plane  of  the 
paper  will  be  represented  by  straight  lines  perpendicular  to  A  B,  such 
as  BC  and  DE.  At  the  instant  when  we  begin  to  reckon  time  the 
incident  wave-front  is  supposed  to  be  in  the  position  shown  by  AB, 
and  hence  at  this  moment  the  disturbance  will  have  just  arrived  at 
the  point  A  of  the  plane  surface  MJ,.  From  this  moment,  therefore, 
according  to  HUYGENS'S  idea,  this  disturbed  point  A  is  itself  to  be 
regarded  as  a  centre  of  disturbance,  and  from  it  as  centre  elementary 
hemispherical  waves  are  propagated  not  only  into  the  second  medium 
(b)  but  also  back  into  the  first  medium  (a).  Exactly  the  same  con- 
dition will  be  true  at  this  instant  (t  —  o)  of  every  point  in  the  plane 
surface  situated  on  the  straight  line  perpendicular  to  the  plane  of  the 
paper  at  the  point  A .  The  envelope  of  each  of  these  two  sets  of  equal 
hemispherical  surfaces  will  be  a  semi-cylinder,  whose  axis  is  the  straight 
line  just  mentioned.  A  little  later  the  disturbance  which  was  ini- 
tially at  D  will  reach  the  point  E  in  the  line  juju ;  and  if  va  denotes  the 
speed  with  which  the  disturbance  is  propagated  in  the  medium  (a) ,  the 
moment  when  it  arrives  at  E  will  be  t  =  DE/va.  Beginning  from 
this  moment  the  two  sets  of  semi-cylindrical  surfaces  which  have  for 
their  common  axis  the  straight  line  perpendicular  at  E  to  the  plane 
of  the  paper  will  begin  to  be  formed.  And,  thus,  at  successively  later 
and  later  instants,  the  disturbance  will  arrive  in  turn  at  all  the  points 
in  nfjL  which  lie  between  A  and  C;  until,  finally,  at  the  time  t  =  BCJva 
the,  disturbance  reaches  the  extreme  point  C.  Meanwhile,  around  all 
the  straight  lines  perpendicular  to  the  plane  of  the  diagram  at  the 
points  on  JJL/JL  which  lie  between  A  and  C  two  sets  of  co-axial  semi- 
cylindrical  elementary  wave-surfaces  have  been  forming,  one  set  being 
propagated  back  into  the  first  medium  (a)  and  the  other  set  being 
propagated  forward  into  the  second  medium  (b).  The  nearer  one  of 


18  Geometrical  Optics,  Chapter  I.  [  §  21. 

these  points  between  A  and  C  is  to  the  point  C,  the  smaller  will  be  the 
radius  of  the  corresponding  semi-cylinder. 

20.  Let  us  consider,  first,  the  Reflected  Wave.     At  the  moment  t  = 
BC/va,  when  the  point  C  begins  to  be  disturbed,  the  semi-cylindrical 
wave  5X  whose  axis  passes  through  A  will  have  expanded  in  the  first 
medium  until  its  radius  is  equal  to  BC.     At  this  same  instant  the 
semi-cylindrical  wave  52  whose  axis  is  determined  by  the  point  E 
will  have  been  expanding  into  the  first  medium  during  the  time  B  C/va 
—  DE/va,  so  that  the  disturbance  will  have  been  propagated  a  dis- 
tance BC  —  DE  =  JC,  which  is  therefore  the  radius  of  this  cylindrical 
surface. 

According  to  HUYGENS'S  Principle,  the  surface  which  at  any  instant 
is  tangent  to  all  the  elementary  semi-cylindrical  reflected  waves  will 
be  the  required  reflected  wave-front  at  that  instant.  We  shall  show 
that  this  reflected  wave-front  is  a  plane  surface  which  at  the  moment 
when  the  disturbance  reaches  C  contains  this  point;  or,  what  amounts 
to  the  same  thing,  we  shall  show  that  if  the  line  CG  in  the  plane  of 
the  diagram  touches  at  G  the  semi-circle  in  which  the  plane  cuts  the 
semi-cylinder  Slt  it  will  be  the  common  tangent  of  all  such  semi- 
circles; for  example,  it  will  be  tangent  to  the  semi-circle  S2  around 
any  point  E  as  centre.  From  C  draw  CG  tangent  to  Sl  at  G  and  CF 
tangent  to  52  at  F.  Draw  AG  and  EF.  The  triangles  CGA  and 
A  B  C  are  congruent,  since  the  angles  at  B  and  G  are  both  right  angles 
and  AG  =  BC.  Hence,  Z  GCA  =  Z  BA  C.  Similarly,  from  the 
congruence  of  the  triangles  CFE  and  CEJ,  it  follows  that  Z  FCE  = 
Z  JEC.  And  since  Z  BA  C  =  Z  JEC,  we  have  Z  GCA  =  Z  FCE] 
and,  consequently,  the  tangent-lines  CG  and  CF  coincide.  Hence, 
the  trace  in  the  plane  of  the  paper  of  the  reflected  wave-front  is  the 
straight  line  CFG.  This  reflected  plane  wave  will  be  propagated 
onwards,  parallel  with  itself,  in  the  direction  shown  in  the  diagram 
by  the  reflected  rays  AG,  EF,  etc.  It  is  evident  from  the  construc- 
tion that  the  ray  incident  at  A ,  the  normal  A  N  to  the  reflecting  sur- 
face at  A  and  the  corresponding  reflected  ray  AG  are  all  situated  in 
the  same  plane,  viz.,  here  the  plane  of  the  paper  which  is  the  plane 
of  incidence  for  the  ray  in  question.  It  only  remains  therefore  to 
show  that  the  angles  of  incidence  and  reflexion  are  equal.  This  is 
obvious  also  from  the  congruence  of  the  triangles  CGA  and  CBA. 

21.  The  Refracted  Wave.     If  the  velocity  of  propagation  of  the 
wave  in  the  second  medium  (b)  is  denoted  similarly  by  vb,  it  is  plain 
that  at  the   moment  /  =  B  C/va  when    the  disturbance   reaches    the 
point  C,  the  secondary  disturbance  which  proceeds   from  A  as  centre 


§  22.]  Fundamental  Laws  of  Geometrical  Optics.  19 

will  have  been  propagated  into  the  medium  (b)  to  a  distance  A  H  = 
vbt  =  vb-  BC/va;  and,  similarly,  the  disturbance  at  any  intermediate 
point,  as  E,  between  A  and  C,  will  have  been  propagated  in  the  second 
medium  to  a  distance  EK  =  (BC  -  DE)vb/va  =  EJ-vb/va.  Thus, 
the  radii  of  the  elementary  semi-cylindrical  refracted  waves  S[  and 
£2,  whose  axes  are  perpendicular  to  the  plane  of  the  paper  at  A  and 
E,  are  BC-vb/vaand  EJ-vbfva,  respectively.  The  refracted  wave- 
front  at  any  instant  will  be  the  surface  which  is  tangent  to  all  these 
elementary  cylindrical  surfaces  at  this  instant.  Exactly  the  same 
method  as  we  used  in  finding  the  reflected  wave-front  can  be  employed 
here;  and  we  shall  find  that  at  the  instant  when  the  disturbance 
reaches  C  the  refracted  wave-front  is  the  plane  containing  the  point 
C  which  is  perpendicular  to  the  plane  of  the  paper  and  tangent  to 
the  elementary  wave  S[  at  H. 

SNELL'S  law  of  refraction  may  be  deduced  at  once  by  observing 
that  in  the  figure  A  G  =  A  C  •  sin  a,  where  a  =  Z.  A  B  C  is  equal  to  the 
angle  of  incidence  of  the  parallel  incident  rays,  and  A  H=A  C-sin  a', 
where  a'  =  Z  A  CH  is  equal  to  the  angle  of  refraction  of  the  parallel 
refracted  rays;  and,  consequently: 

sin  a       AG       va  . 

7  =  -r^>  =  -  =  constant.  (3) 

sin  a!       AH      vb 

22.  In  the  figure  the  case  is  represented  where  the  disturbance  is 
propagated  faster  in  the  first  medium  (a)  than  in  the  second  medium 
(6),  that  is,  va  is  greater  than  vb.  In  this  case  the  angle  of  refraction 
a'  is  less  than  the  angle  of  incidence  a,  and  hence  the  refracted  rays 
are  bent  towards  the  normal,  as,  for  example,  when  light  is  refracted 
from  air  into  glass.  According  to  the  Wave-Theory  of  Light,  there- 
fore, the  velocity  of  propagation  in  the  optically  denser  of  the  two 
media  is  less  than  it  is  in  the  other  medium.  Now  the  NEWTONian  or 
Emission  Theory  of  Light  leads  to  precisely  the  opposite  conclusion. 
The  two  theories  are  here  in  direct  conflict  with  each  other,  and  ex- 
periment has  decided  in  favor  of  the  Wave  Theory.  ARAGO,  in  1838, 
suggested  the  method  of  measuring  the  speed  of  propagation  of  light 
which  was  afterwards  (1865)  successfully  employed  by  FOUCAULT. 
FOUCAULT'S  experiments  demonstrated  that  light  travelled  faster,  for 
example,  in  air  than  in  water.  These  experiments  were  subsequently 
repeated  by  MICHELSON,  with  an  improved  form  of  apparatus,  and 
MICHELSON  found  that  the  speed  of  light  in  air  was  1.33  times  as 
great  as  that  in  water,  which  agrees  with  the  value  of  the  relative 


20  Geometrical  Optics,  Chapter  I.  [  §  24. 

index  of  refraction  of  air  and  water.  The  same  experimenter  found 
that  the  speed  in  air  was  1.77  times  the  speed  in  carbon  bisulphide, 
whereas  the  value  of  n  for  these  two  substances  is  about  1.63,  so  that 
in  this  case  the  agreement  was  not  so  close. 

ART.  7.     ABSOLUTE  INDEX  OF  REFRACTION  OF  AN  OPTICAL  MEDIUM. 

23.  According  to  the  Wave-Theory,  therefore,  the  relative  index 
of  refraction  of  two  media  (a)  and  (b)  is  equal  to  the  ratio  of  the  speeds 
of  propagation  of  light  in  the  two  media.  And,  hence,  if  we  know  the 
indices  of  refraction  of  a  medium  (c)  with  respect  to  each  of  two  media 
(a)  and  (b),  we  can  easily  compute  the  value  of  the  relative  index  of 
refraction  of  the  two  media  (a)  and  (b)  with  respect  to  each  other. 
For,  according  to  formulae  (i)  and  (3),  we  shall  obtain: 


-V  %,  =  ->: 


and  therefore: 


which,  according  to  (2),  may  be  written  also: 


For  example,  suppose  that  the  substances  designated  by  the  letters  a,  b 
and  c  are  water,  glass  and  air,  respectively,  and  that  we  know  the 
values  of  the  relative  indices  of  air  and  water  and  of  air  and  glass,  viz., 
nca  —  4/3  and  ncb  =  3/2;  then  the  value  of  the  relative  index  of  refrac- 
tion from  water  to  glass  will  be  nab  =  (3/2):  (4/3)  =  9/8. 

Generally,  it  may  be  shown  that  if  the  letters  a,  6,  c,  ...  i,  j,  k  are 
employed  to  designate  a  number  of  optical  media,  then: 

nab '  nbc  -ncd--  n{j  -  njk  =  nak.  (4) 

And,  in  particular,  if  the  last  medium  (k)  is  identical  with  the  first 
medium  (a) ,  the  continued  product  of  the  relative  indices  of  refraction 
will  be  equal  to  unity;  formula  (2)  states  this  law  for  the  case  where 
there  are  only  two  media  (a)  and  (b). 

24.  The  fact  that  nab  =  ncb :  nca  suggests  the  idea  of  employing 
some  standard  optical  medium  (c)  with  respect  to  which  the  indices  of 
refraction  of  all  other  media  could  be  expressed.  The  medium  that 


§  25.]  Fundamental  Laws  of  Geometrical  Optics.  21 

is  selected  for  this  purpose  is  that  of  empty  space  or  vacuum,  and  the 
index  of  refraction  of  a  medium  with  respect  to  empty  space  is  called, 
therefore,  the  absolute  index  of  refraction  of  the  medium,  or,  simply, 
the  refractive  index  of  the  medium.  Accordingly,  the  absolute  index 
of  refraction  of  empty  space  is  itself  equal  to  unity,  and  if  na,  nb  denote 
the  absolute  indices  of  two  media  (a)  and  (6),  then  evidently: 


The  absolute  indices  of  refraction  of  all  known  transparent  media 
are  greater  than  unity.  However,  KuNDT1  determined,  in  1888,  the 
indices  of  refraction  of  a  number  of  metallic  substances,  using  very 
thin  prisms  of  the  materials  which  he  subjected  to  investigation;  and 
the  values  of  n  which  he  obtained  in  the  case  of  silver,  gold  and  copper 
were  all  less  than  unity :  which  implies  that  light  travels  faster  in  each 
of  these  metals  than  it  does  in  vacua.  See  also  more  recent  experi- 
ments with  such  substances  as  these,  especially  those  of  DRUDE  and 
MINOR  in  1903. 

The  index  of  refraction  of  air,  at  o°  C.  and  under  a  pressure  of 
76  cm.  of  mercury  for  light  corresponding  to  the  FRAUNHOFER  D-line 
has  been  found  to  be  equal  to  1 .000293  5  ft  *s  usually  taken  as  equal 
to  unity. 

25.  With  every  isotropic  optical  medium  there  is  associated,  there- 
fore, a  certain  numerical  constant  n;  and  thus  when  a  ray  of  light  is 
refracted  from  a  medium  of  index  n  into  another  medium  of  index  n', 
the  trigonometric  formula  of  the  law  of  refraction  may  be  written  in 
the  following  symmetrical  form: 

n  •  sin  a  =  n'  •  sin  a' ;  (6) 

which  may  be  stated  by  saying: 

At  every  refraction  of  a  ray  of  light  from  one  medium  to  another,  the 
product  of  the  refractive  index  of  the  medium  and  the  sine  of  the  acute 
angle  between  the  ray  and  the  incidence-normal  remains  unchanged. 

This  product 

K  =  n  •  sin  a  (7) 

is  sometimes  called  the  Optical  Invariant. 

1  A.  KUNDT:  Ueber  die  Brechungsexponenten  der  Metalle:  Ann.  der  Phys.  (3),  xxxiv. 
(1888),  469-489. 


22  Geometrical  Optics,  Chapter  I.  [  §  27. 

26.  Reflexion  considered  as  a  Special  Case  of  Refraction.  Whereas 
the  angles  of  incidence  and  refraction  have  like  signs  always,  on  the 
contrary  the  signs  of  the  angles  of  incidence  and  reflexion  are  always 
opposite.  In  order,  therefore,  that  formula  (6)  may  be  applicable  also 
to  the  case  of  reflexion  as  well  as  to  that  of  refraction,  the  values  of 
n  and  nr  in  the  former  case  must  be  such  that  a'  =  —  a  is  a  solution 
of  the  equation  in  question;  and  the  condition  that  we  shall  have 
this  solution  is  evidently: 

n'  =  —  n,     or     nf /n  =  —  i. 

Thus,  it  will  not  be  necessary  to  investigate  separately  and  independ- 
ently each  problem  of  reflexion;  for  so  soon  as  we  have  discovered  in 
any  special  case  the  relation  between  the  incident  ray  and  the  corre- 
sponding refracted  ray,  we  have  merely  to  impose  the  condition 


in  order  to  ascertain  directly  the  relation  which  under  the  same  cir- 
cumstances exists  between  the  incident  ray  and  the  corresponding 
reflected  ray.  This  procedure,  which  will  be  frequently  employed  in 
the  following  pages,  will  be  found  to  be  exceedingly  convenient  and 
serviceable,  besides  saving  much  needless  labour. 

Here",  also,  we  take  occasion  to  say  that  hereafter  whenever  we 
speak  of  the  "direction  of  a  straight  line" — that  is,  the  positive  direc- 
tion of  the  line — we  shall  mean  always  the  direction  from  a  point  on 
the  line  in  the  medium  of  the  incident  rays  towards  the  point  where  the 
line  meets  the  reflecting  or  refracting  surface.  If  the  straight  line  is  itself 
the  path  of  an  incident  or  refracted  ray  of  light,  the  positive  direction 
as  thus  defined  will  be  the  direction  along  the  line  in  which  the  light  goes; 
but  if  the  straight  line  is  the  path  of  a  reflected  ray,  the  positive  direc- 
tion in  this  case  (assuming  that  there  is  only  one  reflecting  surface) 
will  be  opposite  to  that  which  the  light  actually  follows.  It  will  be 
well  to  bear  this  in  mind,  especially  in  deriving  reflexion-formulae  from 
the  corresponding  refraction-formulse  by  the  method  above  mentioned., 
(See  §176;  see  also  §251.) 

ART.  8.     THE  CASE  OF  TOTAL  REFLEXION. 

27.     The  formula 

/       n    • 
sin  a  —  —  sin  a 

enables  us  to  calculate  the  magnitude  of  the  angle  of  refraction  a1 ', 


§  27.]  Fundamental  Laws  of  Geometrical  Optics.  23 

so  soon  as  we  know  the  values  of  the  indices  n,  nr  of  the  two  media 
and  the  magnitude  of  the  angle  of  incidence  «;  and  thus  we  can  de- 
termine the  direction  of  the  refracted  ray  corresponding  to  a  given 
incident  ray.  However,  the  solution  of  the  above  equation  is  not  al- 
ways possible,  for  if  the  magnitudes  denoted  by  the  symbols  n,  n'  and 
a  are  such  that  the  expression  on  the  right-hand  side  of  the  equation 
turns  out  to  have  a  value  greater  than  unity,  evidently  there  will  be 
no  angle  a'  that  can  satisfy  the  equation,  and  hence  in  such  a  case 
there  will  be  no  refracted  ray  corresponding  to  the  given  incident  ray. 
In  order  to  make  this  matter  clear,  let  us  distinguish  here  two  cases 
as  follows: 

(1)  The  case  when  n'  >  n;  as,  for  example,  when  the  light  is  re- 
fracted from  air  to  water   (n'/n  =  4/3).     In  this  case  the  second 
medium  is  said  to  be  more  highly  refracting,  or  "optically  denser", 
than  the  first  medium.     The  angle  of  incidence  a  will  be  greater  than 
the  angle  of  refraction  a! ,  so  that  a  ray,  entering  the  second  medium 
from  the  first,  will  be  bent  towards  the  incidence-normal.     Under  these 
circumstances,  the  value  of  the  expression  on  the  right-hand  side  of 
the  above  equation  will  be  always  less  than  unity,  so  that  there  is 
always  a  certain  angle  a!  whose  sine  has  this  value.     Provided  the  second 
medium  is  optically  denser  than  the  first,  to  every  incident  ray  there  will 
always  be  a  corresponding  refracted  ray. 

(2)  The  case  when  n'  <  n\   as,  for  example,  when  the  light  is  re- 
fracted from  water  to  air  (n'/n  =  3/4) ;  in  which  case  the  first  medium 
is  the  optically  denser  of  the  two.     The  angle  of  incidence  a  now  will 
be  less  than  the  angle  of  refraction  a/,  so  that  the  refracted  ray 
will   be    bent   away  from   the  incidence-normal.     When  n  is  greater 
than  n't  the  expression  on  the  right-hand  side  of  the  above  equation 
may  be  less  than,  equal  to  or  greater  than  unity,  depending  on  the 
value  of  the  incidence-angle  a.     For  a  certain  limiting  value  a  =  A 
of  the  angle  of  incidence,  we  shall  have  n-sina/n'  =  i,  and  hence 
a!  =  90°.     In  this  case,  therefore,  the  refracted  ray  corresponding  to 
an  incident  ray  which  meets  the  refracting  surface  at  an  angle  of  in- 
cidence A  such  that 

/ 

sin  A  =  -  =  nab,  (8) 

will  lie  in  the  tangent-plane  to  the  refracting  surface  at  the  point  of 
incidence.  If  the  two  media  (a)  and  (6)  are  separated  by  a  plane 
surface,  the  refracted  ray  in  this  limiting  case  will  proceed  along  the 
surface,  or,  as  we  say,  just  "graze"  the  surface.  This  angle  A  be- 


24 


Geometrical  Optics,  Chapter  I. 


[§27. 


tween  the  incidence-normal  and  the  direction  of  the  ray  in  the  denser 
of  the  two  media  is  called  the  critical  angle  for  the  two  media  (a)  and 
(&).  In  formula  (8)  n  denotes  always  the  refractive  index  of  the  den- 
ser of  the  two  media  (n^  <  i) ;  so  that,  for  example,  if  the  two  media 

are  air  and  water,  the  water 
corresponds  to  medium  (a) 
and  the  air  to  medium  (Z>), 
and  hence  we  have  nab 
=  3/4,  for  which  we  find 
A  =  48°  27'  40".  For  air 
and  glass,  nab  =  2/3  and 
A  =  42°  37'- 

In  this  case  (n  >  n') ,  if  the 
incident  ray  meets  the  re- 

FlG>  7<  fracting  surface  at  an  angle 

TOTAL  INTERNAL  REFLEXION.  °f  incidence  a  greater  than 

the    critical    angle  A,   the 

expression  n  -  sin  a/nf  will  be  greater  than  unity,  which  means  that 
there  will  be  no  real  value  belonging  to  the  angle  a',  and  hence  to  such 
an  incident  ray  there  will  be  no  corresponding  refracted  ray.  The 
ether-disturbance  propagated  in  the  denser  medium  in  such  a  direc- 
tion as  this  will  not  cross  the  boundary-surface  between  the  two  media, 
but  will  be  totally  reflected  there.  Consider,  for  example,  the  diagram 
(Fig.  7),  where  the  point  designated  by  S  represents  a  point-source 
of  light  supposed  to  be  situated  in  a  medium  (a)  which  is  optically 
denser  than  the  medium  (6)  from  which  it  is  separated  by  a  plane 
refracting  surface,  the  trace  of  which  in  the  plane  of  the  paper  is  the 
straight  line  MM-  Rays  are  emitted  from  S  in  all  directions,  but  only 
those  rays  are  refracted  into  the  rarer  medium  (&)  that  are  comprised 
within  the  conical  surface  whose  vertex  is  at  S,  whose  axis  is  the  per- 
pendicular SA  let  fall  from  5  on  /z/j,  and  whose  semi-angle  is  Z  ASB 


=  Z  A  =  sin" 


The  ray  SB  is  refracted  along  the  plane  refract- 


ing surface  in  the  direction  B\i,  as  shown  by  the  arrow-head ;  whereas 
a  ray  SR  which  has  an  angle  of  incidence  a  greater  than  the  critical 
angle  A  is  not  refracted  at  all. 

Another  way  of  regarding  this  diagram  is  to  suppose  that  an  eye 
were  placed  at  the  point  5,  and  that  the  rays  were  being  refracted 
from  the  medium  (6)  into  the  denser  medium  (a) ;  so  that  in  this  case 
the  directions  of  the  arrow-heads  on  the  rays  in  the  figure  should  all 
be  reversed.  All  the  rays  entering  the  eye  at  S  will  be  comprised 
within  the  cone  generated  by  revolving  the  right  triangle  5^4  B  around 


§28.] 


Fundamental  Laws  of  Geometrical  Optics. 


25 


SA  as  axis.  For  example,  suppose  that  the  media  (a)  and  (6)  are 
water  and  air,  respectively,  so  that  W  represents,  therefore,  the  hori- 
zontal free  surface  of  tranquil  water,  and  suppose  that  5  marks  the 
position  below  the  water  of  the  eye  of  an  observer.  An  object  situ- 
ated on  the  horizon  (determined  by  the  water-surface)  would  be  made 
visible  by  means  of  the  ray  BS,  and  the  eye  under  water  would  lo- 
cate the  object  as  being  in  the  air  in  the  direction  SB.  A  ray  coming 
from  a  star  and  falling  on  the  surface  of  the  water  between  A  and  B 
might  enter  the  eye  at  5,  but  the  apparent  zenith-distance  of  the  star 
would  always  be  less  than  its  actual  zenith-distance,  except  when  the 
star  was  actually  at  the  zenith-point  of  the  celestial  sphere. 

The  phenomenon  of  total  reflexion  of  light  at  the  boundary-surface 
between  water  and  air  is  beautifully  exhibited  in  the  luminous  foun- 
tains and  cascades  that  in  recent  years  have  been  spectacular  features 
at  expositions  and  places  of  amusement. 

Incidentally,  it  may  be  remarked  here  that  the  ratio  of  the  inten- 
sity of  the  reflected  light  to  that  of  the  refracted  light  increases  stead- 
ily with  increase  of  the  angle  of  incidence,  from  the  least  value  of  this 
angle  when  the  rays  are 
normally  incident  to  its 
greatest  value  when  the 
rays  are  totally  reflected. 
The  rays  that  are  totally 
reflected  from  the  inside  of 
one  of  the  faces  of  an  equi- 
lateral triangular  glass 
prism  placed  in  the  sun- 
light are  seen  at  a  glance 
to  be  brighter  than  the 
rays  reflected  at  the  outside 
face  of  the  prism. 


ART.  9.     GEOMETRICAL  CON- 
STRUCTIONS, ETC. 


FIG.  8. 


CONSTRUCTION  OF  REFLECTED  AND  REFRACTED  RAYS. 
The  straight  line  rr  is  the  trace  in  the  plane  of  the  paper 
of  the  tangent-plane  at  the  incidence-point  B  to  the 
reflecting  or  refracting  surface.  BP'  =  n'  •  BPln  ; 
PA  =  AP" ;  PB,  BR  and  BQ  represent  the  paths  of  the 
incident,  reflected  and  refracted  rays,  respectively. 


28.  Construction  of  the 
Reflected  Ray.  In  the  dia- 
gram (Fig.  8)  the  straight 
line  PB  represents  the  path 
of  an  incident  ray  meeting  a  reflecting  surface  at  the  incidence-point 
B,  and  N  N'  represents  the  normal  to  this  surface  at  B\  so  that,  if  a 
denotes  the  angle  of  incidence,  Z  NBP  =  a.  The  straight  line  per- 


26  Geometrical  Optics,  Chapter  I.  [  §  29. 

pendicular  to  N  N'  at  the  point  B  is  the  trace  in  the  plane  of  inci- 
dence of  the  tangent-plane  rr  to  the  reflecting  surface  at  B.  In  order 
to  construct  the  corresponding  reflected  ray,  we  draw  from  any  point 
P  of  the  incident  ray  the  straight  line  PA  perpendicular  to  rr  at  A , 
and  prolong  this  perpendicular  to  P"  until  AP"  =  PA,  and  from  P" 
draw  the  straight  line  P"BR',  then  BR  will  represent  the  path  of 
the  corresponding  reflected  ray.  The  proof  of  the  construction  is 
obvious  from  the  figure,  since  we  have: 

Z.NBR  =  LPP'R  =  Z.BPP"  =  £PBN  =  -  a; 

according  to  the  law  of  reflexion. 

29.  Construction  of  the  Refracted  Ray.  Let  n,  n'  denote  the  ab- 
solute indices  of  refraction  of  the  two  isotropic  media  separated  by 
a  smooth  refracting  surface,  and  let  B  (Fig.  8)  designate  the  point 
where  the  given  incident  ray  PB  meets  this  surface.  With  the  inci- 
dence-point B  as  centre,  and  with  any  radius  r  =  BP  describe  in  the 
plane  of  incidence  the  arc  of  a  circle  cutting  the  incident  ray  in  a  point 
P;  and  in  the  same  plane  describe  also  the  arc  of  a  concentric  circle 
of  radius  equal  to  n'r/n.  Through  P  draw  a  straight  line  perpendicular 
at  A  to  the  plane  rr  which  is  tangent  to  the  refracting  surface  at  the 
incidence-point  B ;  and  let  the  straight  line  A  P,  produced  if  necessary, 
meet  the  circumference  of  the  latter  circle  in  a  point  P'  lying  on  the 
same  side  of  the  tangent-plane  as  the  point  P.  Through  the  point 
B  draw  the  straight  line  P1 BQ.  Then  BQ  will  represent  the  path 
of  the  corresponding  refracted  ray.  For 

sinZ  APB       BP'      n' 
sin  Z.AP'B~  BP   ~  n  ' 

and,  since  Z  APB  =  Z  NPB  =  a,  it  follows  from  the  law  of  refrac- 
tion that  Z  AP'B  =  Z  N'BQ  =  a',  where  a'  denotes  the  angle  of 
refraction. 

The  diagram,  as  drawn,  exhibits  the  case  when  the  ray  is  refracted 
into  a  denser  medium  (nf  >  n) ;  but  the  construction  given  above  is 
equally  applicable  to  the  other  case  also. 

Assuming  that  n'  >  n,  we  see  from  Fig.  8  that  when  the  angle  of 
incidence  Z  NBP  =  90°,  the  incident  ray  PB  will  be  tangent  to  the 
refracting  surface  at  the  incidence-point  J5,  and  then  BA  =  BP,  so 
that  AP'  will  be  tangent  to  the  construction-circle  of  radius  B  P. 
In  this  case  we  shall  have: 

P7?  w 

a'  =  Z  PP'B  =  sin-1 -=7=  =  sin"1-  =  A, 
r  D  n 


§30.] 


Fundamental  Laws  of  Geometrical  Optics. 


27 


where  A  denotes  the  magnitude  of  the  so-called  critical  angle  of  the 
two  media  (§  27). 

30.  The  Deviation  of  the  Refracted  Ray.  The  angle  between  the 
directions  of  the  incident  and  refracted  rays  is  called  the  angle  of  devi- 
ation, and  will  be  denoted  here  by  the  symbol  e.  Thus,  if  PB  (Fig.  9) 
represents  the  path  of  a  ray  incident  on  a  refracting  surface  at  the 
point  3,  and  if  P'B  (constructed  as  explained 
in  §  29)  shows  the  direction  of  the  correspond- 
ing  refracted  ray,  then  Z  P' BP  =  e  ;  that  is, 
e  denotes  the  acute  angle  through  which  the 
direction  of  the  refracted  ray  has  to  be  turned 
to  bring  it  into  the  same  direction  as  that  of 
the  incident  ray.  If  a  =  Z  NBP  =  Z.APB, 
a.'  =  Z  NBP'  =  /.A P'B  denote  the  angles 
of  incidence  and  refraction,  the  angle  of  devi- 
ation is  denned  by  the  following  relation: 


=  a  —  a 


(9) 


FIG.  9. 

DEVIATION  OF  THE  RE- 
FRACTED RAY.  The  straight 
lines  PB,  P'B  show  the  direct- 


LAP'B 


LP'PB 


The  diagram  is  drawn  for  the  case  when  jons  °f  the  incident  and  re- 

»•"••'•       -i  .  r  t«       fractedrays. 

n  >  n,  for  which  the  sign  of  the  angle  e  is 
positive.  By  merely  interchanging  the  letters 
P,  P'  in  the  figure,  we  obtain  the  case  when 
n'  <  n,  for  which  the  angle  denoted  by  c  is  negative. 

It  is  apparent  from  the  figure  that  the  intercept  P' P  included  be- 
tween the  circumferences  of  the  two  construction-circles,  which  re- 
mains always  parallel  to  the  normal  BN,  increases  in  length  as  the 
angle  of  incidence  a  increases;  and,  since  the  other  two  sides  BP  and 
BP'  of  the  triangle  BPP'  have  constant  lengths,  it  follows  that  the 
deviation  of  the  refracted  ray  increases  with  increase  of  the  angle  of  in- 
cidence. This  is  true  both  for  n'  >  n  and  for  nf  <  n. 

Differentiating  equation  (6),  we  obtain  (after  eliminating  n,  n'): 


da.'  _  tana' 
da        tan  a. 


(10) 


and,  since  from  the  figure  tan  a'/tan  a  =  AP/AP',  we  have  there- 
fore the  following  relations: 

da':da:de  =  AP:  AP' :PP' ', 
so  that  in  the  triangle  PP'B  the  side  PP'  opposite  the  angle  e  is 


28  Geometrical  Optics,  Chapter  I.  [  §  32. 

divided  externally  at  A  into  segments  which  are  inversely  propor- 
tional to  the  corresponding  variations  of  the  angles  at  P  and  P' '. 
Moreover,  since 

de  _  PP' i 

da~  AP'~  I  +  AP/PP" 

and  since  as  the  angle  a  increases,  not  only  does  AP  decrease  but  PP' 
increases  by  an  equal  amount,  it  follows  that  de/da  increases  with 
increase  of  a.  Hence, 

The  greater  the  angle  of  incidence,  the  greater  will  be  the  corresponding 
rate  of  increase  of  the  angle  of  deviation. 

This  characteristic  property  of  refraction  is  true  both  for  n'  /n 
greater  than  unity  and  for  n'/n  less  than  unity.  In  the  case  of  re- 
flexion, the  law  will  be  different,  for  the  deviation  of  the  reflected 
ray  decreases  in  proportion  as  the  angle  of  incidence  increases. 

ART.  10.     CERTAIN  THEOREMS  CONCERNING  THE  CASE  OF  SO-CALLED 
OBLIQUE  REFRACTION  (OR  REFLEXION). 

31.  The  plane  of  incidence  containing  the  normal  to  the  refracting 
(or  reflecting)  surface  at  the  point  of  incidence  is  a  normal  section  of 
the  surface  at  that  point;  and,  whenever  feasible,  it  will  be  conven- 
ient to  select  this  plane  as  the  plane  of  the  diagram.     In  the  following 
pages,  however,  we  shall  often  have  occasion  to  investigate  the  path  of 
a  ray  which  is  incident  in  succession  on  a  series  of  refracting  (or  re- 
flecting) surfaces ;  in  which  case  the  plane  of  incidence  with  respect  to 
one  such  surface  will,  in  general,  make  with  the  plane  of  incidence  with 
respect  to  the  next  following  surface  an  angle  different  from  zero. 
Accordingly,  in  our  diagrams  it  may  happen  that  the  normal  section 
of  the  refracting  (or  reflecting)  surface  which  lies  in  the  plane  of  the 
paper  may  not  coincide  with  the  normal  section  which  contains  the 
ray  incident  on  that  surface  and  its  corresponding  refracted  (or  re- 
flected) ray.     In  such  a  case  as  this  the  ray  is  said  to  be  "obliquely" 
incident  on  the  surface ;  and  in  this  connection  the  following  theorems 
will  be  found  useful. 

We  remark  that  it  will  be  necessary  to  treat  here  only  the  problem 
of  refraction;  as  the  corresponding  theorems  relating  to  reflexion, 
which  may  easily  be  proved  independently  also,  may  be  derived  im- 
mediately by  merely  putting  n'  —  —  n,  according  to  the  general  prin- 
ciple explained  in  §  26. 

32.  In  the  diagram  (Fig.  10)  the  straight  line  N  N'  is  the  normal 
at  the  point  B  to  the  refracting  surface,  so  that  the  plane  of  the  dia- 


§32.] 


Fundamental  Laws  of  Geometrical  Optics. 


29 


gram  is  therefore  the  plane  of  a  normal  section  of  the  surface  with 
respect  to  the  point  B',  and  the  straight  line  rr  is  the  line  of  inter- 
section of  the  plane  of  the 
paper  with  the  plane  tan- 
gent to  the  refracting  sur- 
face at  B.  Let  RB  rep- 
resent the  path  of  a  ray 
incident  on  the  surface  at 
the  point  B,  and  from  any 
point  R  of  this  ray  draw 
R  N  perpendicular  at  N  to 
the  normal  NN'.  The 
plane  of  incidence  RB  N  is 
also  the  plane  of  a  normal 
section  of  the  surface;  but 
we  shall  suppose  here  that 
R  B  is  "obliquely"  incident 
on  the  refracting  surface, 
so  that  the  plane  of  inci- 
dence is  not  the  same  as  the 
plane  of  the  paper.  The 
corresponding  refracted  ray 
BR'  will  lie  in  the  plane  of  incidence, 
point  R',  such  that 

RBiBR'  = 


FIG.  10. 

OBLIQUE  REFRACTION.  RB,  BR'  represent  the  paths  of 
the  incident  and  refracted  rays.  £  NBR  =  a,  L  N'BR' 
=  «',  Z  RBP  =  r),  L  R' BP'  =  >?',  Z  NBP  =  y,  £  N'BP' 

-y. 


On  this  refracted  ray  take  a 


and  from  R'  draw  R' N'  perpendicular  to  N  N'  at  N'.  Draw  also 
RP,  R'P'  perpendicular  to  the  plane  of  the  paper.  Then  the  two 
planes  RPN,  R' Pf  N'  will  be  parallel  to  the  tangent-plane  at  B.  By 
the  law  of  refraction: 

n  -  sin  a.  =  n'  •  sin  a', 


where    Z  NBR  =  a,    Z  N'BR'  =  a'.     By  the  construction: 
RN  =  RB-  sin  a,  R'N'  =  R'B  •  sin  a' ; 
RN  =  N'R'. 


and,  therefore: 


Since  RN  and  N'R',  lying  both  in  the  plane  of  incidence,  are  equal 
and  parallel,  PN  and  N1 P' ',  which  are  the  projections  of  RN  and 
N'R'  in  the  plane  of  the  paper,  are  also  equal  and  parallel ;  so  that  the 
triangles  NPR  and  N'P'R'  are  congruent,  and  RP  =  P'R'. 


30  Geometrical  Optics,  Chapter  I.  [  §  33. 

If  the  symbols  77,  77'  are  employed  to  denote  the  angles  made  by  the 
incident  and  refracted  rays  RB,  BRf  with  their  projections  PB,  BPf 
in  the  plane  of  the  normal  section  which  is  the  plane  of  the  paper, 
so  that 

Z  RBP  =  77,      Z  R'BP'  =  77', 
then,  since 

RP  =  RB-sini),     R'P'  =  R'B  •  sin  17', 
we  have: 

RB-sini)  =  £.R'-sm77'; 
and,  hence: 

n-sinrj  =  n'  -sin??'.  (ll) 

Accordingly,  we  have  the  following  result: 

The  sines  of  the  angles  which  the  incident  and  refracted  rays  make 
with  the  plane  of  any  normal  section  of  the  refracting  surface  at  the  point 
of  incidence  have  the  same  ratio  as  the  sines  of  the  angles  of  incidence  and 
refraction  themselves. 

33.     Moreover,  since,  by  construction  (Fig.  10), 


n'-RB  = 
and 

PB  =  RB-  cos  77,     BP'  =  BR'  •  cos  77', 
we  have: 

n'-PB-cosr)  =  n  •  BP'  •  cos  77'. 
Putting 

Z  NBP  =  7,      Z  N'BP'  =  7', 
so  that 

PB  -  sin  7  =  BP'  •  sin  7', 
we  find: 

n  -cos  77  -sin  7  =  n'  •  cos  77'  -sin  7';  (12) 

a  result  which  may  be  stated  as  follows: 

Tine  projections  of  the  incident  and  refracted  rays  on  a  plane  of  a  nor- 
mal section  of  the  refracting  surface  at  the  point  of  incidence  are  also 
subject  to  a  law  of  refraction,  the  absolute  indices  of  refraction  n  -  cos  77 
and  n'  -  cos  77'  being  dependent  on  the  angles  77  and  77'  made  by  the  inci- 
dent and  refracted  rays  with  the  plane  of  the  normal  section. 

If  we  put 

n1J  =  n-cosirjJ     nj  =  n'  •  cos  y'  , 

and  bear  in  mind  that  we  have  also  the  relation: 

n-sin  v\  =  n'  •  sin  77', 


§  34.]  Fundamental  Laws  of  Geometrical  Optics, 

we  can  derive  easily  the  formula: 


31 


in  the  form  given  by  CORNU/ 

34.  The  following  is  a  convenient  method  of  constructing  a  draw- 
ing representing  the  path  of  a  ray  obliquely  refracted  at  the  surface 
of  separation  of  two  isotropic  optical  media. 

Let  the  plane  of  the  paper  (Fig.  n)  be  designated  as  the  xy- plane 
and  let  the  tangent-plane  to  the  refracting  surface  at  the  incidence- 
point  B  be  designated  as  the  yz- plane,  which  is  represented  as  making 


FIG.  11. 
CONSTRUCTION  OF  OBLIQUELY  REFRACTED  RAY. 

an  acute  angle  with  the  plane  of  the  paper.  From  B,  in  a  plane  xz 
perpendicular  to  the  plane  of  the  paper,  draw  B  N  normal  at  B  to 
the  tangent-plane  yz,  and  draw  BM  normal  at  M  to  the  plane  of  the 
paper.  Suppose,  for  example,  that  the  real  length  of  BM  is  twice  its 
length  as  shown  in  the  figure.  Let  P  designate  the  position  of  the 
point  where  the  given  incident  ray  meets  the  plane  of  the  paper,  so 
that  BP  shows  the  direction  of  the  incident  ray  lying  in  the  plane  of 
incidence  BP  N.  If  the  triangle  B  P  N  is  revolved  around  P  N  as 
axis  until  it  comes  into  the  plane  of  the  paper,  the  point  B  will  arrive 
at  a  point  C  on  the  straight  line  drawn  from  M  perpendicular  to  NP, 

1A.  CORNU:  De  la  refraction  a  travers  un  prisme  suivant  une  loi  quelconque:  Ann. 
ec.  norm.  (2),  i.  (1872),  237.  See  also  E.  REUSCH:  DieLehre  von  der  Brechung  u.Far- 
benzerstreuung  des  Lichts  an  ebenen  Flaechen  und  in  Prismen,  die  in  mehr  synthetischer 
Form  dargestellt:  POGG.  Ann.  cxvii.  (1862),  247;  and  A.  BRAVAIS  :  Notice  sur  les 
parhelies  qui  sont  situes  a  la  meme  hauteur  que  le  soleil:  Journ.  ic.  polyt.,  xviii.,  cah.  30 
(1845),  79;  and  Memoire  sur  les  halos:  Journ.  ec.  polyt.,  xviii.,  cah.  31  (1847),  27. 


32  Geometrical  Optics,  Chapter  I.  [  §  34. 

and  Z  NCP  =  a.  Hence,  with  C  as  centre  and  with  radius  equal 
to  n'  -  CP\n  describe  in  the  plane  of  the  paper  the  arc  of  a  circle 
meeting  in  a  point  R  the  straight  line  drawn  through  P  parallel  to 
CN;  evidently,  as  in  §29,  Z  NCR  =  a! .  Therefore,  the  straight  line 
BQ  joining  the  incidence-point  B  with  the  point  Q  where  CR  meets 
NP  will  represent  in  the  diagram  the  direction  of  the  refracted  ray. 


CHAPTER  II. 

CHARACTERISTIC   PROPERTIES  OF  RAYS  OF  LIGHT. 
ART.  11.     THE  PRINCIPLE  OF  LEAST  TIME  (LAW  OF  FERMAT). 

35.  FERMAT1  (1608-1665),  arguing  from  an  assumed  law  of  the 
economy  of  nature  that  light  must  be  propagated  from  one  point  to 
another  in  the  shortest  time,  was  able  to  deduce  the  law  of  refraction 
in  the  case  of  a  ray  refracted  from  one  isotropic  medium  to  another 
across  a  plane  boundary-surface;  or,  conversely,  that  the  time  required 
by  the  light  to  be  transmitted  from  any  point  P  on  the  incident  ray 
to  any  point  Q  on  the  corresponding  refracted  ray  is  less  than  it  would 
be  along  any  other  route  between  the  points  P  and  Q.     A  correspond- 
ing law  in  regard  to  light  reflected  at  a  plane  mirror  dates  back  to 
HERO  of  Alexandria  (150  B.  C.). 

If  the  boundary-surface  separating  the  two  media  is  curved,  the  time 
taken  by  the  light  to  be  transmitted  from  P  to  Q  along  the  actual 
path  may  not,  however,  be  always  a  minimum;  on  the  contrary,  in  cer- 
tain cases  it  may  be  a  maximum.  A  simple  illustration  is  given  by 
Sir  WM.  ROWAN  HAMILTON,2  who  instances  the  fact  that  "if  an  eye 
is  placed  in  the  interior,  but  not  at  the  centre,  of  a  reflecting  hollow 
sphere,  it  may  see  itself  reflected  in  two  opposite  points,  of  which  one 
indeed  is  the  nearest  to  it,  but  the  other  on  the  contrary  is  the  farthest; 
so  that  of  the  two  different  paths  of  light,  corresponding  to  these  two 
opposite  points,  the  one  indeed  is  the  shortest,  but  the  other  is  the 
longest  of  any." 

36.  A  characteristic  property  of  a  ray  of  light  may  be  stated  quite 
generally  as  follows: 

//  a  ray  of  light,  undergoing  any  number  of  reflexions  and  refractions, 
connects  two  points  P  and  Q,  the  time  taken  by  the  light  to  be  transmitted 
from  P  to  Q  along  the  actual  path  of  the  ray  is  either  a  minimum  or  a 
maximum. 

It  will  be  entirely  sufficient  if  we  prove  the  truth  of  this  statement 
merely  for  the  case  of  a  single  refraction;  as  it  can  then  be  extended 
immediately  to  the  case  where  the  ray  suffers  any  number  of  reflexions 
and  refractions. 

1  P.  FERMAT:  Litterce  ad  P.  MERSENUM  contra  Dioptricam  Cartesianam  (Paris,  1667). 
2W.  R.  HAMILTON:  On  a  General  Method  of  expressing  the  Paths  of  Light  and  of 
the  Planets:  Dublin  University  Review,  October,  1833. 

4  33 


34 


Geometrical  Optics,  Chapter  II. 


[§36. 


N 


In  the  diagram  (Fig.  12)  the  point  designated  by  P  represents  the 
starting  point  in  the  first  medium  and  the  point  designated  by  Q 
represents  the  terminal  point  in  the  second  medium;  and  w  is  the 
trace  of  the  refracting  surface  in  a  plane  containing  the  two  points  P 

and  Q  (which  is  represented  here  as  the 
plane  of  the  paper).  The  problem  to 
be  solved  is,  What  must  be  the  position 
on  the  refracting  surface  of  the  incidence- 
point  B,  in  order  that  the  time  taken 
by  the  light  to  be  transmitted  from  P 
to  Q,  viz., 

PB  ,  BQ 


where  v  and  vr  denote  the  speeds  of 
propagation  of  light  in  the  first  and 
second  medium,  respectively,  shall  be 
either  a  minimum  or  a  maximum?  Let 
us  suppose  that  this  point  B  is  also  situ- 
ated in  the  plane  of  the  paper,  which 
will  be  therefore  the  plane  of  incidence 
of  the  ray  PB.  Evidently,  this  critical 
position  of  the  incidence-point  B  will 
be  such  that  an  infinitely  small  variation  from  this  position  would 
not  alter  the  time  taken  by  the  light  in  going  from  P  to  Q.  It  will 
suffice  to  consider  a  variation  of  the  position  of  B  in  the  plane  of  in- 
cidence; accordingly,  let  us  designate  by  Bl  the  position  of  a  point 
on  the  normal  section  w  of  the  refracting  surface  infinitely  close  to 
the  critical  point  of  incidence  B.  If  the  light  travelled  from  P  to  Q 
along  the  route  PB&,  the  time  taken  would  be: 


FIG.  12. 

FERMAT'S  I,AW  OF  I<EAST  TIME. 
PB,  BQ  represent  paths  of  incident 
and  refracted  rays.  PB\Q  is  another 
hypothetical  route  from  the  point  P 
in  the  first  medium  to  the  point  Q  in 
the  second  medium,  which  differs  infi- 
nitesimally  f rom  the  actual  route  PBQ. 


and,  consequently,  the  condition  which  has  to  be  imposed  is  that 

PB  -  PB,      BQ  -  B,Q  =  Q 

v  vf 


From  B  and  Bl  draw  BRl  and 
at  Rl  and  at  R,  respectively;  then 


perpendicular  to  B&  and  PB 


and 


§  38.]  Characteristic  Properties  of  Rays  of  Light.  35 

and,  hence,  the  condition  above  becomes: 

KB  _  B.R,  = 
v          v' 

Draw    NBN'  normal  to  the  curve  MM  at  the  point   B,  and  put 
Z  NBP  =  a,  /.  N'BQ  =  a';  then 

RB  =  BB1  -  sin  a,     B^  =  BBl  -  sin  a'. 
The  condition  may  be  written,  therefore: 

sin  a 


which  will  be  recognized  as  the  law  of  refraction  (§21).  But  the  actual 
path  of  the  ray  from  P  to  Q  is  according  to  this  law.  Consequently, 
the  time  t  along  this  path  will  be  either  a  minimum  or  a  maximum. 

Whether  in  any  given  special  case  the  time  is  a  minimum  or  a  maxi- 
mum, can  be  determined  only  by  investigating  the  form  of  the  re- 
fracting (or  reflecting)  surface. 

37.  This  result,  as  was  stated  above,  can  be  immediately  extended 
to  the  case  where  the  ray  is  compelled  in  its  progress  from  P  to  Q 
to  traverse  any  number  of  media  or  to  bend  away  at  certain  surfaces 
of  separation  between  two  bodies,  that  is,  where  the  ray  is  constrained 
to  undergo  a  certain  prescribed  series  both  of  refractions  and  of  re- 
flexions. If  we  denote  by  tk  the  time  occupied  by  the  light  between 
two  successive  adventures  of  this  kind,  the  analytical  expression  of 
the  so-called  Principle  of  Least  Time  may  be  written  in  the  following 
form: 

=  o;  (13) 


that  is,  the  time  taken  by  the  light  to  be  transmitted,  under  certain 
prescribed  conditions,  from  one  point  P  to  another  point  Q  along  the 
actual  path  of  the  ray  differs  from  the  time  which  would  be  taken 
along  any  other  hypothetical  route,  which  is  infinitely  near  to  the 
actual  route,  by  an  infinitesimal  of  an  order  higher  than  the  first  order. 
38.  The  Optical  Length  of  a  Ray;  and  the  Principle  of  the  Short- 
est Route.  The  sum  of  the  products  of  the  length  of  the  path  of  a 
ray  in  each  medium  by  the  refractive  index  of  that  medium  is  called 
the  Optical  Length,  sometimes  also  the  reduced  length,  of  the  ray.  Thus 
if  /!,  /2,  etc.,  denote  the  actual  lengths  of  the  ray-path  in  the  media 


36  Geometrical  Optics,  Chapter  II.  [  §  39. 

whose  indices  of  refraction  are  denoted  by  nlt  nz,  etc.,  respectively, 
the  optical  length  of  the  ray  is  : 


where  nkJ  lk  denote  the  values  of  the  magnitudes  n,  I  for  the  kth 
medium.  When  the  ray  is  reflected  at  a  body  i,  we  must  put  here 
ni  =  —  %_!,  according  to  the  rule  given  in  §  26;  so  that  the  definition 
given  above  applies  to  reflexions  as  well  as  to  refractions. 

Since  nk  =  V/vk,  where  V  and  vk  denote  the  speeds  of  propagation 
of  light  in  vacuo  and  in  the  kth  medium  of  the  series,  respectively, 
and  lk  =  Vjfk1  we  have  nhlk  =  Vtk;  whence  we  see  that  the  optical 
length  of  the  ray  (  =  F-S/ft)  is  equal  to  the  distance  that  light  would 
travel  in  vacuo  in  the  same  length  of  time  as  it  takes  to  go  over  its 
actual  path.  This  explains  the  use  of  the  term  "reduced  length". 

We  also  see  that  equation  (13)  is  equivalent  to  the  following: 

d(2nklk)  =  o;  (14) 

whence  is  derived  the  so-called  Principle  of  the  Shortest  Route,  which 
may  be  stated  as  follows: 

When  light  is  transmitted  from  one  point  P  to  another  point  Q,  under- 
going during  its  progress  any  prescribed  series  of  reflexions  and  refrac- 
tions, the  optical  length  measured  along  the  actual  path  of  the  ray  is  a 
minimum  or  a  maximum. 

ART.  12.     HAMILTON'S  CHARACTERISTIC  FUNCTION. 

39.  The  statement  at  the  end  of  the  last  article  recalls  MAUPER- 
TUIS'S  celebrated  "Principle  of  Least  (or  Stationary)  Action",  after- 
wards developed  by  EULER  and  other  great  mathematicians  ;  provided 
we  define  the  vague  term  "action"  in  the  case  of  a  ray  of  light  to 
mean  the  optical  length  of  the  ray.  The  function 


is  the  so-called  Characteristic  Function,  the  idea  of  which  was  first 
introduced  into  mathematical  optics  by  Sir  W.  R.  HAMILTON,  and 
which  reduces  the  solution  of  all  problems,  in  theory  at  least,  to  one 
common  process.1 

1  Professor  P.  G.  TAIT  in  his  book  on  Light  (Edinburgh,  1889)  says  (Art.  189):  "  HAMIL- 
TON was  in  possession  of  the  germs  of  this  grand  theory  some  years  before  1824,  but  it  was 
first  communicated  to  the  Royal  Irish  Academy  in  that  year,  and  published  in  imperfect 
instalments  some  years  later."  HAMILTON'S  papers  on  this  subject  published  under  the 
title  "  Theory  of  systems  of  rays  "  are  to  be  found  in  the  Transactions  of  the  Royal  Irish 
Academy,  xv.  (1828),  69-174;  xvi.  (1830),  3-62;  and  93-126;  and  xvii.  (1837),  I-I44- 


§  40.]  Characteristic  Properties  of  Rays  of  Light.  37 

In  the  application  of  this  method  the  co-ordinates  of  the  two  ter- 
minal points  P(a,  b,  c)  and  Q(a',  &',  c')  which  are  connected  by  the  ray 
are  to  be  regarded  as  known,  and,  therefore,  invariable.  The  equa- 
tions of  the  reflecting  and  refracting  surfaces  must  likewise  be  given. 
But  the  co-ordinates  of  the  points  where  the  ray  meets  these  surfaces 
are  the  variables  in  the  problem.  The  equation  of  the  kth  surface 
may  be  written: 


and  since  the  co-ordinates  xk,  yk,  zk  of  the  point  where  the  ray  meets 
this  surface  must  satisfy  this  equation,  we  may  regard  zk  as  a  known 
function  of  xk,  yk.  The  actual  length  of  the  ray-path  between  the 
(k  —  i)th  and  the  kth  surfaces  will  be: 


Since  dL  =  o,  we  must  have: 


dL  dL 

T-  =  o,     —  =  o, 
dxk  dyk 


where  zk  is  to  be  considered  as  the  dependent  variable,  so  that: 


dxk      dxk       dzk  dxk '     dyk     dyk       dzk  dyk ' 

40.  In  order  to  illustrate  the  use  of  the  method  in  a  simple  case, 
let  us  suppose  that  there  is  only  one  refracting  surface  separating  two 
media  of  refractive  indices  n  and  ri r;  then: 

h  =  I  =  y/(xl-a)2+(yi-b)2+(zl-c)\ 
and 

and,  according  to  the  equations  above,  we  derive  here: 

(  ^  -i_  t          \  ^zi  /  /         \    ,    /  /         >.  dzv 

jj         \xl      a)  -j-  ^f,      c;  ,  (a  —  x^  -\-  (c  —  gj 


dxl 

dL         (^-«  +  (%—  >  (b'~yi)  + 


38  Geometrical  Optics,  Chapter  II.  [ 

If  the  incidence-point  is  taken  as  the  origin  of  co-ordinates,  then 
xl  =  yl  =  zl  =  o.  Moreover,  if  the  incidence-normal  is  taken  as  the 
z-axis,  then  also  dzljdxl  =  dzl/dyl  =  o.  Introducing  these  simpli- 
fying values,  we  find: 

na      n'a'  nb      rib' 

~T~    ~Y  =0>    J'-T   =0< 

If,  further,  we  take  the  plane  of  incidence  for  the  ys-plane,  we  must 
put  a  =  o ;  whence  it  follows  from  the  first  of  the  two  equations  above 
that  a'  =  o  also;  and  hence  the  point  Q,  and  therefore  also  the  re- 
fracted ray,  must  lie  in  the  plane  of  incidence,  in  accordance  with  a 
fundamental  law  of  refraction.  Finally,  if  a,  a'  denote  the  angles  of 
incidence  and  refraction,  it  is  evident  that: 

b  V 

y  =  —  sin  a,     jr  =  sin  a  , 

and  hence  the  second  equation  above  is  equivalent  to  the  other  fun- 
damental law  of  refraction: 

n-  sin  a  =  ri  •  sin  a' '. 

Thus  we  see  how  this  process  leads  to  the  ordinary  laws  of  refraction. 
41.  If  the  characteristic  function  of  a  system  is  known,  it  is  pos- 
sible in  theory  to  deduce  from  it  all  the  optical  properties  of  the  sys- 
tem. In  some  comparatively  simple  cases  this  process  enables  us  to 
get  results  with  almost  magical  facility.  It  must  be  admitted,  how- 
ever, that  the  method,  so  fascinating  on  account  of  its  generality,  is 
difficult  in  its  applications,  involving  as  it  does  the  theories  of  the 
higher  analytical  geometry  and  demanding  mathematical  knowledge 
and  skill  of  the  highest  order.  In  addition  to  HAMILTON,  a  number 
of  other  investigators,  among  whom  may  be  mentioned  especially 
MAXWELL1  and  TniESEN2  and  BRUNS,S  have  developed  in  one  way  or 

1  J.  C.  MAXWELL:  A  dynamical  theory  of  the  electromagnetic  field,  Proc.  Roy,  Soc., 
xiii.  (1864),  531-536;  Phil.  Trans.,  civ.  (1865),  459-512;  Phil.  Mag.,  (4)  xxix.  (1865), 
152-157.  Also,  On  the  application  of  HAMILTON'S  characteristic  function  to  the  theory 
of  an  optical  instrument  symmetrical  about  an  axis:  Proc.  of  London  Math.  Soc.,  vi. 
(1874-5),  117-122;  and  On  HAMILTON'S  characteristic  function  for  a  narrow  beam  of 
light;  Proc.  London  Math.  Soc.,  vi.  (i874~'5),  182-190. 

3  M.  THIESEN:  Beitraege  zur  Dioptrik:  Berl.  Ber.,  1890,  799-813.  Also,  Ueber  voll- 
kommene  Diopter,  WIED.  Ann.  der  Phys.  (2),  xlv.  (1892),  82i-'3.  See  also,  Ueber  die 
Construction  von  Dioptern  mit  gegebenen  Eigenschaften,  WIED.  Ann.  der  Phys.  (2), 
xlv.,  823-'4. 

3  H.  BRUNS:  Das  Eikonal:  Saechs.  Ber.  d.  Wiss.,  xxi.  (1895),  321-436.  See  also  F. 
KLEIN:  Ueber  das  BRUNSche  Eikonal;  and,  also,  Raeumliche  Kollineation  bei  optischen 
Instrumenten:  Zft.  f.  Math.  u.  Phys.,  xlvi.  (1901). 


§  42.]  Characteristic  Properties  of  Rays  of  Light.  39 

another  the  theory  of  the  characteristic  function  in  optics.  But  the 
greatest  difficulty  is  encountered  in  turning  the  theory  to  account, 
and,  so  far  as  the  practical  optician  is  concerned,  the  HAMiLTONian 
method  has  not  been  found  to  smooth  his  way. 

ART.  13.     THE  LAW  OF  MALUS. 

42.  The  wave-front  at  any  instant  due  to  a  disturbance  emanat- 
ing from  a  point-source  is  the  surface  which  contains  all  the  farthest 
points  to  which  the  disturbance  has  been  propagated  at  that  instant. 
Thus,  the  wave-surface  may  be  denned  as  the  totality  of  all  those 
points  which  are  reached  in  a  given  time  by  a  disturbance  originating 
at  a  point.  In  a  single  isotropic  medium  the  wave-surfaces  will  be 
concentric  spheres  described  around  the  point-source  as  centre;  but 
if  the  wave-front  arrives  at  a  reflecting  or  refracting  surface  ju>  at 
which  the  directions  of  the  so-called  rays  of  light  are  changed,  the  form 
of  the  wave-surface  thereafter  will,  in  general,  be  spherical  no  longer; 
and  even  in  those  cases  when  the  refracted  (or  reflected)  wave-front 
is  spherical,  the  centre  (except  under  certain  very  special  circum- 
stances) will  not  coincide  with  the  centre  of  the  incident  wave-surfaces. 
The  function  1>ril  (§38)  has  the  same  value  for  all  actual  ray-paths 
between  one  position  of  the  wave-surface  and  another  position  of  it; 
so  that  knowing  the  form  of  the  wave-front  at  any  instant  and  the 
paths  of  the  rays,  we  may  construct  the  wave-front  at  any  succeeding 
instant  by  laying  off  equal  optical  lengths  along  the  path  of  each  ray. 
It  follows  that  the  ray  is  always  normal  to  the  wave-surface.  For, 
suppose  that  the  straight  line  PB  represents  the  path  of  a  ray  inci- 
dent at  the  point  B  on  a  surface  ju  separating  two  media,  and  that  the 
straight  line  BQ  represents  the  path  of  the  corresponding  refracted 
(or  reflected)  ray;  and  let  a  designate  the  wave-surface  whereon  the 
point  Q  lies.  From  the  incidence-point  B  draw  any  straight  line  BR 
meeting  the  wave-surface  a  in  the  point  designated  by  R.  Then,  by 
the  minimum  property  of  the  light-path,  the  route  PBQ  is  less  than 
the  route  PBR,  because  the  natural  route  from  P  to  R  is  not  via 
the  incidence-point  B;  and  hence  the  straight  line  BQ  must  be  shorter 
than  the  straight  line  BR,  and  therefore  BQ  is  the  shortest  line  that 
can  be  drawn  from  the  incidence-point  B  to  the  wave-surface  a.  It 
follows  that  BQ  meets  the  wave-surface  a  normally.  The  same  rea- 
soning will  be  applicable  also  in  the  case  of  every  other  refraction  or 
reflexion,  so  that  we  may  state  generally: 

The  light-rays  meet  the  wave-surface  normally,  and,  conversely,  the 
system  of  surfaces  which  intersect  at  right  angles  the  rays  emanating 
originally  from  a  point-centre  is  a  system  of  wave-surfaces. 


40  Geometrical  Optics,  Chapter  II.  [  §  44. 

43.  The  fact  which  has  just  been  proved  is  equivalent  to  the  law 
enunciated   by  MALUS/    in  1808,  which  may  be  stated  as  follows: 

An  orthotomic  system  of  rays  remains  orthotomic,  no  matter  what  re- 
fractions (or  reflexions')  the  rays  may  undergo  in  traversing  a  series  of 
isotropic  media.  (An  orthotomic  system  of  rays  is  one  for  which  a 
surface  can  be  constructed  which  will  cut  all  the  rays  at  right  angles.) 
A  proof  of  this  law  which  does  not  contain  any  reference  to  the 
ideas  of  the  Wave-Theory  is  given  by  HEATH2  as  follows: 

Let  ABODE  (Fig.  13)  and  A1B1C1D1E1  be  two  infinitely  near  ray- 
paths,  and  suppose  that  they  cross  nor- 
mally at  A  and  Al  a  certain  surface  a.  On 
each  ray  of  the  system,  reckoning  from  the 
points  A,  Alt  etc.,  where  the  rays  cross  the 
surface  cr,  let  a  series  of  points  E,  Elt  etc., 
be  determined  such  that  the  optical  lengths 
from  A  to  E,  from  Al  to  Elt  etc.,  are  all 
equal.  We  propose  to  show  that  the  surface 
a'  which  contains  the  terminal  points  £,  Elt 
13  etc.,  of  these  rays  will  cut  the  rays  at  right 

I,AW  OF  MALUS.  angles. 

In  order   to   prove  this,   we   draw  the 

straight  lines  A^B  and  DEl  as  shown  in  the  figure.  The  optical 
length  2«/  measured  along  the  infinitely  near  hypothetical  route 
AlBCDEl  is,  by  FERMAT'S  Law,  equal  to  2nl  along  A1B1C1D1E1 
or  along  ABCDE.  Hence,  subtracting  from  each  the  part  BCD 
which  is  common  to  the  routes  ABCDE  and  A^BCDE^  we  have: 

n-AB-\-n'-DE  =  n-  Afi  +  n'-DEl9 

where  n,  nr  denote  the  refractive  indices  of  the  first  and  last  medium, 
respectively.  But  since  AB  is  normal  to  the  surface  o-,  ultimately 
AiB  =  AB-,  and,  hence,  ultimately  also  DE  =  DE^  that  is,  DE 
must  be  normal  to  the  surface  cr'.  In  the  same  way  we  can  show 
that  any  other  ray  D±E±  will  likewise  be  normal  to  o-'. 

ART.  14.     OPTICAL  IMAGES. 

44.  In  case  we  do  not  wish  to  utilize  all  the  rays  emitted  from  a 
luminous  body,  we  may  interpose  a  screen  with  a  suitable  opening 
in  it,  whereby  some  of  the  rays  are  intercepted,  while  others,  called 

1  E.  L.  MALUS:  Optique:  Journ.  de  I'Ecole  Polyt.,  vii.  (1808),  1-445  84-129. 
«R.  S.  HEATH:  A  Treatise  on  Geometrical  Optics  (Cambridge,  1887),  Art.  87. 


§  45.]  Characteristic  Properties  of  Rays  of  Light.  41 

the  ''effective  rays",  are  permitted  to  pass  through  the  opening.  Thus, 
each  separate  point  of  a  luminous  body  is  to  be  regarded  as  the  vertex 
of  a  cone  or  bundle  of  rays.  In  every  bundle  of  rays  there  is  always  a 
certain  central  or  representative  ray,  usually  coinciding  with  the  axis 
of  the  cone,  or  distinguished  in  some  special  way,  called  the  chief  ray1 
of  the  bundle.  A  pencil  of  rays  is  obtained  from  a  bundle  of  rays  by 
passing  a  plane  through  the  axis  or  chief  ray  of  a  bundle.  This  use 
of  this  term  is  convenient  and  is  also  in  accordance  with  the  usage 
of  some  writers  on  geometry. 

An  optical  system  is  a  combination  of  isotropic  media  arranged  in 
a  certain  sequence  so  that  they  are  traversed  by  the  effective  rays 
all  in  the  same  order.  In  this  case  the  effective  rays  emitted  by  a 
luminous  point  P  are  those  rays  coming  from  P  which  succeed  finally 
in  passing  through  the  system  from  one  end  to  the  other  without  being 
intercepted  at  any  point  on  the  way.  In  general,  through  any  point 
P',  within  the  region  reached  by  the  bundle  of  emergent  rays  which 
had  their  origin  at  the  luminous  point  P,  one  ray,  and  one  ray  only, 
will  pass,  since  the  optical  route  between  P  and  P',  for  a  given  dis- 
position of  the  optical  media,  will  usually  be  uniquely  determined. 
However,  within  this  region  there  may  be  found  a  number  of  points 
P'  where  two  or  more  rays  intersect ;  and  under  certain  circumstances 
it  may  indeed  happen  that  all  of  the  effective  rays  emanating  from 
the  point  P  will,  after  traversing  the  optical  system,  meet  again  in 
one  point  P']  and  in  this  exceptional  case  the  point  P'  is  said  to  be 
the  optical  image  of  the  point  P,  and  the  two  points  P  and  P',  object- 
point  and  image-point,  are  called  conjugate  points  or  conjugate  foci. 
If  the  rays  actually  pass  through  P',  the  image  is  said  to  be  real; 
whereas  if  it  is  necessary  to  produce  backwards  the  actual  portions  of 
the  rays  in  order  to  make  them  intersect  in  P',  the  image  is  said  to 
be  virtual.  Thus,  in  the  case  of  a  perfect  image,  all  of  the  "emer- 
gent rays"  corresponding  to  the  rays  of  a  given  bundle  of  "incident 
rays"  proceeding  from  the  object-point  P  will  intersect  in  the  image- 
point  P'. 

45.  In  order,  therefore,  to  have  an  image  in  the  sense  above  de- 
fined, the  optical  system  must  transform  a  train  of  spherical  waves 
with  the  object-point  P  as  centre  into  another  train  of  spherical  waves 
with  the  image-point  P'  as  centre.  The  optical  lengths  along  all  the 
ray-paths  between  P  and  P'  will  be  equal,  so  that  the  disturbances 

1  The  term  "  chief  ray  "  is  a  happy  rendering  of  the  German  Hauptstrahl  which  has 
been  introduced  into  English  Optics  by  Professor  SILVANUS  P.  THOMPSON  in  his  transla- 
tion of  Dr.  O.  LUMMER'S  Photographic  Optics  (London,  1900). 


42  Geometrical  Optics,  Chapter  II.  [  §  46. 

arrive  at  P'  along  all  these  different  routes  all  in  the  same  phase,  and 
hence  conspire  to  produce  at  P'  a  maximum  effect.  According  to 
the  notions  of  Geometrical  Optics,  there  will  be  no  light-effects  what- 
ever at  points  which  lie  outside  of  the  cone  of  rays  which  meet  in  P' ; 
but  when  the  matter  is  investigated  by  the  surer  methods  of  Physical 
Optics,  we  discover  that  this  conclusion  is  not  justified,  and  that  there 
are  light-effects  at  points  which  are  not  comprised  within  this  geo- 
metric cone.  In  fact,  instead  of  a  single  image-point  P',  we  find  that 
we  have  around  P'  a  so-called  diffraction-pattern.  But  the  wider  the 
cone  of  rays  that  meet  in  Pf ,  the  more  nearly  will  the  distribution 
of  light  around  P'  approach  as  its  limit  the  ideal  image-point  of  Geo- 
metrical Optics;  and  this  is  the  only  meaning  which  Physical  Optics 
can  attach  to  the  idea  of  an  image-point. 

ART.  15.  CHARACTER  OF  AN  INFINITELY  NARROW  BUNDLE  OF  OPTICAL  RAYS. 

46.  Caustic  Surfaces.  According  to  the  Law  of  MALUS,  the  direc- 
tion of  the  ray-path  at  any  point  P  is  along  the  normal  to  the  wave- 
surface  which  passes  through  P.  In  the  special  case  when  the  wave- 
surface  is  spherical,  the  normals  all  meet  in  one  point  at  the  centre  of 
the  sphere;  but  if  the  wave-surface  has  any  other  form,  a  pair  of  nor- 
mals drawn  to  the  surface  at  two  different  points  will,  in  general, 
not  intersect  at  all.  The  curved  line  which  is  traced  on  the  surface 
by  a  plane  containing  the  normal  to  the  surface  at  the  point  P  is 
called  a  normal  section  of  the  surface  at  this  point.  The  curvatures 
of  these  lines  at  the  point  P  will  generally  be  different  for  different 
normal  sections ;  and  EULER  has  shown  that  at  each  point  P  of  a  curved 
surface  the  normal  sections  of  maximum  and  minimum  curvature  are 
at  right  angles  to  each  other;  and,  accordingly,  the  two  normal  sections 
thus  distinguished  are  called  the  Principal  Sections  of  the  surface  at 
the  point  P. 

An  investigation  of  the  theory  of  the  curvature  of  surfaces  shows 
that  the  normals  at  consecutive  points  of  a  curved  surface  will  intersect 
each  other,  provided  those  points  are  taken  along  the  curves  of  greatest 
and  least  curvatures;  but  that,  in  general,  the  normals  at  consecutive 
points  do  not  intersect. 

Applying  these  results  from  the  theory  of  curved  surfaces,  let  us 
designate  by  the  symbol  u  the  chief  ray  of  an  infinitely  narrow  bundle 
of  rays,  and  let  P  designate  the  position  of  the  point  on  the  wave- 
surface  a  where  the  chief  ray  u  crosses  this  surface.  Only  those  rays 
of  the  elementary  bundle  which  cross  the  wave-surface  cr  at  the  points 
lying  in  the  principal  sections  of  the  surface  through  the  point  P  will 


§  46.]  Characteristic  Properties  of  Rays  of  Light.  43 

meet  the  chief  ray  u\  so  that  this  ray  u  is  to  be  regarded  also  as  the 
chief  ray  of  each  of  two  infinitely  narrow  pencils  of  rays  lying  in  two 
perpendicular  planes:  the  vertices  of  these  two  pencils  of  rays  being 
the  centres  of  greatest  and  least  curvature  of  the  surface  with  respect 
to  the  point  P.  The  other  rays  of  the  infinitely  narrow  bundle  which 
do  not  lie  in  the  planes  of  the  principal  sections  will  generally  not  meet 
the  chief  ray  u  at  all.  Thus,  on  each  ray  u  determined  by  a  point 
P  of  the  wave-surface,  there  are  to  be  found  two  points,  the  centres 
of  greatest  and  least  curvature  with  respect  to  the  point  P,  which  are 
the  vertices  of  two  narrow  pencils  of  consecutive  rays  of  which  u  is  the 
chief  ray. 

A  line  of  curvature  is  a  curve  traced  on  a  surface  such  that  the  nor- 
mals at  any  two  consecutive  points  of  the  curve  intersect  each  other. 
Therefore,  through  every  ordinary  point  of  the  surface  two  such  lines 
of  curvature  will  pass  intersecting  each  other  at  right  angles.  The 
totality  of  each  of  these  two  systems  of  lines  of  curvature  completely 
covers  the  entire  surface.  The  locus  of  the  points  of  intersection  of 
rays  belonging  to  points  which  lie  along  a  line  of  curvature  will  be 
the  evolute  of  that  line  of  curvature;  and  in  optics  this  evolute,  which 
is  also  the  envelop  of  the  rays  crossing  the  wave-surface  at  points 
lying  along  the  line  of  curvature,  is  called  a  caustic  curve.  The  total- 
ity of  the  caustic  curves  corresponding  to  one  system  of  lines  of  cur- 
vature of  the  curved  surface  will  constitute  a  caustic  surface.  Thus, 
there  will  be  two  caustic  surfaces,  one  for  each  of  the  two  systems  of 
the  lines  of  curvature  of  the  wave-surface ;  these  caustic  surfaces  being 
indeed  the  loci  of  the  two  centres  of  principal  curvature  of  the  wave- 
surface.  Each  ray  is  evidently  a  common  tangent  of  the  two  caustic 
surfaces. 

In  the  special  case  when  the  wave-surface  is  a  surface  of  revolu- 
tion, so  that  the  orthotomic  system  of  rays  is  therefore  symmetrical 
with  respect  to  the  axis  of  revolution,  it  is  easy  to  obtain  a  clear  idea 
of  the  caustic  surfaces.  For  here  one  system  of  lines  of  curvature 
are  the  meridian  curves  of  the  surface,  and  consequently  the  caustic 
surface  corresponding  thereto  is  generated  by  the  revolution  about  the 
axis  of  symmetry  of  the  evolute  of  the  meridian  curve.  And  the 
other  system  of  lines  of  curvature  are  circles  with  their  centres  ranged 
along  the  axis  of  symmetry,  and,  since  the  rays  which  cross  the  wave- 
surface  at  points  lying  in  the  circumference  of  one  of  these  circles  will 
all  lie  in  the  surface  of  a  right  circular  cone  whose  vertex  is  on  the 
axis  of  revolution,  the  caustic  surface  corresponding  to  this  system  of 
lines  of  curvature  reduces  to  a  segment  of  the  axis  of  revolution  itself. 


44  Geometrical  Optics,  Chapter  II.  [  §  47. 

In  Chapter  VI.  of  HEATH'S  Geometrical  Optics  (Cambridge,  1887) 
the  reader  who  wishes  to  pursue  this  subject  will  find  an  extensive 
investigation  of  the  forms  and  properties  of  caustic  lines  and  surfaces 
in  a  number  of  interesting  special  cases.  WOOD'S  Physical  Optics 
(New  York  and  London,  1905),  wherein  the  caustic  surfaces  are 
studied  especially  from  the  standpoint  of  the  Wave-Theory,  and 
experimentally  rather  than  mathematically,  contains  also  much  on  this 
subject  that  is  both  novel  and  suggestive.  However,  so  far  as  the 
theory  and  design  of  optical  instruments  is  concerned,  it  will  hardly 
repay  us  here  to  attempt  to  investigate  these  surfaces  in  detail; 
although  in  the  next  chapter,  by  way  of  illustration,  we  shall  study 
briefly  the  caustic  in  the  case  of  the  refraction  of  a  spherical  wave 
at  a  plane  surface  (§54). 

47.  The  main  thing  that  it  concerns  us  to  know  at  present  is  that 
a  narrow  bundle  of  optical  rays,  originally  homocentric  (or  monocentric, 
as  it  is  sometimes  called,  that  is,  emanating  all  from  one  and  the  same 
point  or  "focus"),  is,  in  general,  transformed  by  reflexion  or  refraction 
at  a  surface  of  any  form  into  a  non-homocentric  or  astigmatic  bundle 
of  rays,  all  the  rays  of  which,  at  least  to  a  first  approximation,  inter- 
sect two  infinitely  short  image-lines,  the  so-called  image-lines  of  the 
bundle.  We  proceed  to  explain  how  this  occurs,  according  to  the 
theory  of  STURM/  the  originator  of  the  theory  of  astigmatism. 

Let  P  (Fig.  14)  designate  the  position  of  a  point  on  the  wave-sur- 
face a,  and  let  the  ray  u  determined  by  the  point  P  be  represented 
in  the  diagram  by  the  straight  line  PSS.  This  ray  coincides  with  the 
normal  to  the  surface  at  the  point  P,  and  the  points  designated  by 
5,  5  are,  in  the  case  shown  in  the  diagram,  the  centres  of  greatest  and 
least  curvature,  respectively,  with  respect  to  the  point  P.  With  5 
as  centre,  and  with  radius  equal  to  SP,  describe  in  the  plane  of  the 
paper  the  infinitely  small  arc  APB  of  a  circle,  making  AP  =  PB: 

1  J.  C.  STURM:  Memoire  sur  1'optique:  LIOUVILLE'S  Journ.  de  Math.,  iii.  (1838),  357-384- 
Also,  Memoire  sur  la  theorie  de  la  vision:  Comptes  rend.,  xx  (1845),  554-560;  761-767; 
1238-1257.  This  latter  paper  was  translated  and  published  in  POGG.  Ann.,  Ixv.  (1845). 

See  also  the  following  writers  on  this  subject: 

E.  E.  KUMMER:  Allgemeine  Theorie  der  gradlinigen  Strahlensysteme :    CRELLES  Journ., 
Ivii.  (1860),  180-230.     Modelle  der  allgemeinen,  unendlich  duennen,  gradlinigen   Strah- 
lenbuendel:  Berl.  Akad.  Ber.,  1860,  469-474.     Ueber  die  algebraischen  Strahlensysteme, 
in's  Besondere  ueber  die  der  ersten    und  der  zweiten  Ordnung:  Berl.  Akad.  Monatsber. 
1865,  288-293.     Berl.  Akad.  Abh.,  1866,  No.  i,  1-120. 

H.  HELMHOLTZ:  Handbuch  der  physiologischen  Optik,  ii.  Thl.  (1860),  246. 
A.  F.   MQEBIUS:  Geometrische  Entwickelung  der  Eigenschaften  unendlich  duenner 
Strahlenbuendel:  Sitzungsber.  d.  Saechs.  Akad.  Math.-phys.  CL,  xiv.  (1862),  1-16. 

F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme  an 
Kugelflaechen :  Denkschr.  d.  Wien.  Akad.,  math.-phys.  CL,  xxxviii.  (1878),  163-192. 

C.  NEUMANN:  Sitzungsber.  d.  Saechs.  Akad.,  math.-phys.  CL,  1879,  42. 


§  47.]  Characteristic  Properties  of  Rays  of  Light.  45 

and  through  the  point  S  draw  in  the  plane  of  the  paper  the  straight 
line  d~Sdf  perpendicular  to  the  ray  u.  Let  the  figure  thus  obtained 
be  rotated  about  ddf  as  axis  through  an  infinitely  small  arc  above  and 
below  the  plane  of  the  paper,  so  that  the  arc  APE  will  thereby  gen- 


FIG.  14. 

CONSTITUTION  OF  INFINITELY  NARROW,  ASTIGMATIC  BUNDLE  OF  OPTICAL  RAYS.  Aa'p'b'Bbpa 
is  the  element  of  the  wave-surface  <r  around  the  point  P.  PSS  represents  the  normal  to  the  surface 
at  the  point  P,  and  S,  S  designate  the  positions  on  this  normal  of  the  centres  of  principal  curvature 
with  respect  to  the  point  P.  The  chief  ray  u  of  the  astigmatic  bundle  of  rays  is  represented  by  the 
normal  PSS;  S  and  ,5"  are  the  primary  and  secondary  image-points,  respectively  ;  the  infinitely  short 
lines  cc'  and  dd'  are  the  primary  and  secondary  image-lines,  respectively,  cc'  is  perpendicular  to 
the  chief  ray  u  at  the  point  6"  and  lies  in  a  plane  perpendicular  to  the  plane  of  the  paper :  dd'  is 
perpendicular  to  the  chief  ray  u  at  the  point  6"  and  lies  in  the  plane  of  the  paper. 

erate  the  element  of  surface  apb  Bb' 'p' 'a' 'A ,  which,  to  a  first  approx- 
imation, may  be  regarded  as  an  element  da  of  the  wave-surface  a-  in 
the  immediate  vicinity  of  the  point  P.  We  shall  investigate,  there- 
fore, the  infinitely  narrow  bundle  of  rays  which  cross  the  wave-surface 
at  points  lying  within  this  surface-element  da,  of  which  the  ray  u 
proceeding  from  P  is  the  chief  ray. 

During  the  rotation  around  dd'  as  axis  the  point  P  traces  the  in- 
finitely short  arc  pp',  which  is  an  element  of  one  of  the  lines  of  curva- 
ture which  pass  through  P,  the  arc  APB  being  an  element  of  the  other 
line  of  curvature  at  this  point.  The  point  S  traces  the  infinitely  short 
arc  cc'  parallel  to  pp' \  the  arcs  cc'  and  pp'  being  both  perpendicular 
to  the  plane  of  the  paper.  The  rays  proceeding  from  the  points  of 
the  wave-surface  which  lie  along  pp'  constitute  a  narrow  pencil  of 
rays  lying  in  a  plane  perpendicular  to  the  plane  of  the  paper  and  hav- 
ing its  vertex  at  the  point  6*.  And,  similarly,  the  rays  proceeding 


46  Geometrical  Optics,  Chapter  II.  [  §  47. 

from  the  points  of  the  wave-surface  which  lie  along  the  arc  APB  con- 
stitute a  narrow  pencil  of  rays  lying  in  the  plane  of  the  paper  and  hav- 
ing its  vertex  at  the  point  S.  The  chief  ray  u  is  common  to  both  of 
these  pencils. 

The  entire  bundle  of  rays  may  be  regarded  as  composed  of  a  sheaf  of 
pencils  of  rays  in  either  of  two  ways,  as  follows: 

First,  the  entire  bundle  of  rays  may  be  regarded  as  arising  from  the 
rotation  about  ccr  as  axis  of  the  pencil  of  rays  pSp',  so  that  the  element 
of  arc  pp'  generates  the  element  of  surface  da. 

Again,  the  entire  bundle  of  rays  may  be  generated  by  rotating  the 
pencil  of  rays  ASB  about  dd'  as  axis  through  infinitely  small  arcs 
above  and  below  the  plane  of  the  paper. 

The  centres  of  curvature  5,  5,  both  situated  on  the  chief  ray  u,  are 
(as  has  been  stated)  the  so-called  image-points  of  the  infinitely  narrow 
astigmatic  bundle  of  rays.  The  point  S  which  is  the  vertex  of  the 
pencil  of  rays  lying  in  the  plane  of  the  paper  (the  meridian  or  primary 
principal  section  of  the  bundle)  is  called  the  primary  image-point;  and 
the  point  S  which  is  the  vertex  of  the  pencil  of  rays  lying  in  the  plane 
perpendicular  to  the  plane  of  the  paper  (the  sagittal  or  secondary  prin- 
cipal section  of  the  bundle)  is  called  the  secondary  image-point.  In 
the  figure  as  here  shown,  the  point  designated  by  5  is  the  centre  of 
the  greatest  curvature  of  the  surface  with  respect  to  the  point  P,  and 
the  point  designated  by  5  is  the  centre  of  least  curvature;  but  this 
will  depend  entirely  on  the  form  of  the  surface  at  P. 

The  two  infinitely  short  lines  ccf  and  dd' ,  through  both  of  which 
all  the  rays  of  the  bundle  pass,  and  which,  regarded  as  straight  lines, 
lie  in  planes  at  right  angles  to  each  other,  and  which,  moreover, 
are  both  perpendicular  to  the  chief  ray  u  of  the  bundle,  are  the 
two  image-lines  of  the  narrow  astigmatic  bundle  of  rays.  The  primary 
image-line  cSc'  lies  in  the  primary  principal  section,  and  the  secondary 
image-line  dSdf  lies  in  the  secondary  principal  section. 

Thus,  according  to  STURM'S  Theory,  the  general  characteristics 
of  an  infinitely  narrow  bundle  of  optical  rays  may  be  enumerated  as 
follows : 

(1)  The  direction  of  propagation  of  the  disturbance  at  the  point  P 
is  along  the  ray  u  which  is  normal  to  the  wave-surface  a  at  P.     As  a 
first  approximation,  the  element  do-' of  the  wave-surface  at  this  point 
may  be  regarded  as  bounded  by  arcs  which  are  parallel  to  the  arcs 
of  greatest  and  least  curvature  of  the  surface  at  the  point  P. 

(2)  All  rays  of  the  bundle  which  cross  the  wave-surface  at  points 
lying  along  the  arc  APB  (Fig.  14)  intersect  in  the  primary  image- 


§  48.]  Characteristic  Properties  of  Rays  of  Light.  47 

point  S',  whereas  the  rays  which  cross  the  wave-surface  at  points 
lying  along  the  arc  pPp'  intersect  in  the  secondary  image-point  5.  The 
rays  which  cross  the  element  d<r  of  the  wave-surface  at  points  lying 
along  any  arc  drawn  parallel  to  the  arc  APB  will  all  meet  (as  a  first 
approximation)  in  a  point  of  the  primary  image-line  cc' ,  and  such  rays 
will  also  cross  the  plane  of  the  pencil  ASB  (the  primary  principal 
plane)  at  points  which  (to  the  same  degree  of  approximation)  will 
lie  in  the  secondary  image-line  ddf.  Similarly,  rays  which  cross  the 
element  do-  of  the  wave-surface  at  points  which  lie  along  any  arc 
parallel  to  the  arc  pPpf  will  (as  a  first  approximation,  also)  meet  in 
a  point  of  the  secondary  image-line  ddf,  and  will  cross  the  plane  of 
the  pencil  pSp'  (the  secondary  principal  plane)  at  points  which  lie 
in  the  primary  image-line  cc' . 

(3)  If  through  the  normal  u  to  the  wave-surface  at  the  point  P  we 
pass  a  plane  making  with  the  plane  of  the  paper  an  angle  between 
o°  and  90°,  and  consider  the  system  of  rays  which  cross  the  surface- 
element  dff  at  points  lying  along  the  arc  traced  by  this  plane,  we  find 
that  in  general  these  rays  will  not  intersect  each  other  at  all;  for 
ordinarily  this  system  of  rays  will  not  lie  in  the  plane  of  a  normal 
section  of  the  wave-surface. 

The  image-line  at  S  (or  at  5)  contains  the  vertices  of  all  those  pen- 
cils of  rays  which  have  their  planes  perpendicular  to  the  plane  of  prin- 
cipal curvature  for  which  3  (or  5)  is  the  centre.  If  we  are  given 
the  chief  ray  u  and  the  two  image-lines,  we  can  construct  the  entire 
bundle  of  rays  by  joining  each  point  of  one  image-line  with  all  the 
points  of  the  other  image-line. 

48.  With  the  passage  of  time,  the  element  d<r  of  the  wave-front 
advances  in  the  direction  of  the  wave-normal  u,  each  point  of  da- 
travelling  along  the  normal  belonging  to  it.  Approaching  the  image- 
line  at  S,  the  element  shrinks  in  dimensions,  collapsing  finally  at  S 
into  the  image-line  cc' .  Thereafter,  the  surface  element  begins  to  open 
out  again,  and,  later,  it  begins  again  to  contract  until  it  collapses  at 
S  into  the  image-line  ddftt  after  which  the  wave  expands  again  in  a 
sort  of  wedge-shaped  opening.  In  any  position  of  the  element  do- 
lying  between  the  two  image-points  5  and  3,  the  principal  curvatures 
will  necessarily  be  of  opposite  signs,  so  that  while  the  element  will 
be  expanding  along  one  dimension,  it  will  be  contracting  along  the 
other.  At  some  place,  therefore,  between  the  primary  and  secondary 
image-points  a  plane  perpendicular  to  the  chief  ray  will  cut  the  bundle 
of  rays  in  a  section  whose  contour  will  have  a  form  similar  to  that  of 
the  element  do-  of  the  wave-surface.  This  is  the  so-called  place  of 


48  Geometrical  Optics,  Chapter  II.  [  §  49. 

least  confusion.  For  example,  if  the  element  da  is  in  the  form  of  a 
circle,  the  sections  of  the  bundle  of  rays  made  by  planes  at  right  angles 
to  the  axis  of  the  bundle  or  chief  ray  will  generally  be  elliptical  in 
form,  and  at  the  place  of  least  confusion  the  two  axes  of  the  elliptical 
section  will  be  equal,  so  that  we  have  here  a  "circle  of  least  confu- 
sion". Between  the  wave-surface  and  the  place  of  the  circle  of  least 
confusion  the  major  axes  of  the  elliptical  sections  will  be  parallel  to 
the  primary  image-line  cc' \  whereas  the  major  axes  of  the  other  ellip- 
tical sections  beyond  the  circle  of  least  confusion  will  be  parallel  to 
the  secondary  image-line  dd'. 

49.  In  the  above  discussion  it  has  been  assumed  that  the  lines  of 
curvature  at  the  different  points  of  the  element  da-  of  the  wave-surface 
are  parallel  to  the  lines  of  curvature  at  the  point  P;  which  is  true, 
however,  only  in  case  we  neglect  magnitudes  of  the  second  order  of 
smallness.  Hence,  the  results  which  are  given  above  as  to  the  con- 
stitution of  an  infinitely  narrow  bundle  of  optical  rays,  known  as 
STURM'S  Theory,  are  valid  only  to  that  degree  of  approximation.  Tak- 
ing account  of  magnitudes  of  the  second  order  of  smallness,  we  shall 
find  that,  instead  of  image-lines  going  through  the  image-points  of 
the  bundles  of  rays,  we  have  bits  of  image-surfaces,  which,  however, 
in  special  cases  may  collapse  into  image-lines  having  any  inclinations 
to  the  chief  ray.1 

In  the  case  when  the  magnitudes  of  the  second  order  of  small- 
ness  are  neglected,  the  question  has  been  raised,  also,  especially  by 
MATTHIESSEN,2  as  to  the  part  of  STURM'S  Proposition  which  asserts 
that  the  two  image-lines  are  perpendicular  to  the  chief  ray  u.  If,  for 
example,  the  wave-surface  is  a  surface  of  revolution,  and  if  we  draw  two 
infinitely  near  normals  in  the  plane  of  a  meridian  curve,  and  rotate 
the  meridian  plane  through  a  small  angle  about  the  axis  of  revolu- 
tion, we  obtain  for  the  secondary  image-line  the  piece  of  the  axis  inter- 
cepted by  the  two  normals,  which  may  not,  and  generally  will  not, 
be  perpendicular  to  the  chief  normal  u.  The  other  image-line  here 

1  See,  for  example,  LUDWIG  MATTHIESSEN:  Ueber  die  Form  der  unendlich  duennen 
astigmatischen  Strahlenbuendel  und  ueber  die  KuMMER'schen  Modeller  Sitzungber. 
der  math.-phys.  Cl.  der  koenigl.  bayer.  Akad.  der  Wiss.  zu  Muenchen,  xiii.  (1883),  35-51. 

2L.  MATTHIESSEN:  Neue  Untersuchungen  ueber  die  Lage  der  Brennlinien  unendlich 
duenner  copulirter  Strahlenbuendel  gegen  einander  und  gegen  einen  Hauptstrahl:  Ada 
Math.,iv(iS84),  177-192.  Also,  published  in  the  "  Supplement  "  of  Zs.  f.  Math.  u.  Phys., 
xxix.  (1884),  86.  See  also,  by  same  author,  Untersuchungen  ueber  die  Constitution  un- 
endlich duenner  astigmatischer  Strahlenbuendel  nach  ihrer  Brechung  in  einer  krummen 
Oberflaeche:  Zft.  f.  Math.  u.  Phys.,  xxxiii.  (1888),  167-183. 

In  connection  with  this  question,  see  especially  also: 

S.  CZAPSKI:  Zur  Frage  nach  der  Richtung  der  Brennlinien  in  unendlich  duennen  op- 
tischen  Buescheln:  WIED.  Ann.,  xlii.  (1891),  332-337. 


§49.]  Characteristic  Properties  of  Rays  of  Light.  49 

will  be  the  infinitely  small  arc  of  a  circle  described  about  the  axis  by 
the  point  of  intersection  of  the  two  normals. 

In  order  to  make  the  matter  clear,  consider  the  diagram  (Fig.  15), 
where  P  designates  a  point  on  the  wave-surface,  and  where  the  straight 
line  PSS,  drawn  normal  to  the  surface  at  the  point  P,  represents  the 


ct' 


FIG.  15. 
CONSTITUTION  OF  INFINITELY  NARROW,  ASTIGMATIC  BUNDLE  OF  OPTICAL  RAYS. 

chief  ray  u  of  the  bundle.  The  arcs  A  PB  and  pPp'  are  elements  of 
the  lines  of  curvature  of  the  surface  which  go  through  P  and  which 
lie  in  two  planes  at  right  angles  to  each  other.  The  rays  which  cross 
the  wave-surface  at  points  lying  in  the  arc  APB  meet  at  the  centre 
of  curvature  5  of  this  arc ;  and,  similarly,  the  rays  which  cross  the  wave- 
surface  at  points  lying  in  the  arc  pPp'  meet  at  the  centre  of  curvature 
S.  Neglecting  magnitudes  of  the  second  order  of  smallness,  we  may 
consider  the  element  of  the  wave-surface  around  the  point  P  as  a 
curvilinear  rectangle,  with  its  sides  parallel  to  the  arc  APB  and  pPp'. 
The  two  principal  planes  pp'~5  and  ABS  will  be  tangent  to  the  two 
caustic  surfaces  in  the  primary  image-line  cc'  and  in  the  secondary 
image-line  dd' ',  respectively.  Obviously,  as  MATTHIESSEN  contends, 
these  image-lines  may  not  be,  and,  indeed,  generally,  will  not  be,  per- 
pendicular to  the  chief  ray  u.  In  case  the  curvatures  of  the  wave- 
surface  are  symmetrical  on  both  sides  of  pp'  or  of  AB,  or  if  the  cur- 
vature at  the  point  P  is  constant,  the  image-lines  will  be  perpendicu- 
lar to  the  chief  ray.  Thus,  for  example,  if  a  bundle  of  rays  is  re- 
fracted at  a  surface  of  revolution  whose  axis  lies  in  the  same  plane  as 
the  chief  ray  of  the  bundle,  there  will  be  symmetry  on  both  sides  of 
the  arc  pp't  and,  hence,  in  such  a  case  as  this  the  primary  image- 
line  cc'  will  be  perpendicular  to  the  chief  ray  u  at  S,  but  the  secondary 
image-line  dd'  will  not  meet  u  at  3  at  right  angles.  If  the  vertex  of 
the  homoceritric  bundle  of  incident  rays  lies  on  the  axis  of  revolu- 
tion, the  secondary  image-line  will  be  an  element  of  the  axis  of  revo- 
lution containing  the  point  ~S. 

5 


50  Geometrical  Optics,  Chapter  II.  [  §  49. 

Concerning  the  matter  here  under  discussion,  CZAPSKI 's  argument 
(in  his  paper  referred  to  above)  is  substantially  as  follows: 

All  the  rays  of  the  bundle  may  be  regarded  as  intersecting  both  of 
the  image-lines  cc',  ddf,  provided  we  neglect  infinitesimals  of  the  sec- 
ond order.  But  with  this  same  proviso,  we  may  consider  as  image- 
line  any  section  of  the  bundle  of  rays  made  by  a  plane  passing  through 
either  5  or  S.  The  form  of  this  section  will  resemble  more  or  less 
a  figure  8.  The  axes  of  the  two  lemniscate-like  sections  at  5  and  at 
~S  will  be  at  right  angles  to  each  other,  and  these  axes  may  themselves 
be  regarded  as  the  image-lines.  Therefore,  taking  the  sections  normal 
to  the  chief  ray,  we  have,  according  to  this  view,  a  perfect  right  to 
say  that  the  image-lines  are  perpendicular  to  the  chief  ray;  it  being 
merely  a  question  as  to  what  is  meant  by  an  image-line. 

Thus,  the  image-lines  of  the  bundle  of  rays  may  be  defined  in  two 
ways,  and  the  only  question  is  as  to  which  is  to  be  preferred.  Accord- 
ing to  the  first  definition,1  the  image-lines  are  lines  traced  on  the  caustic 
surfaces,  and  as  such  are  distinguished,  therefore,  by  the  following 
properties:  (i)  Each  point  of  them  is  the  "focal"  point  or  meeting- 
place  of  elementary  wave-trains  (or  rays)  of  equal  optical  lengths; 
so  that  assuming  that  these  waves  have  a  common  origin,  they  will 
re-inforce  each  other  at  this  convergence-point  on  the  image-line,  and 
hence  at  this  point  there  will  be  a  maximum  light-effect.  (This  latter 
item  is  not  mentioned  by  MATTHIESSEN,  but  CZAPSKI  directs  atten- 
tion to  it  as  being  a  matter  not  to  be  overlooked  in  this  discussion.) 
And  (2),  as  MATTHIESSEN  very  particularly  remarks,  in  these  lines 
the  section  of  the  bundle  is  a  minimum,  in  some  cases  indeed  an  infini- 
tesimal of  order  higher  than  that  of  any  other  section. 

However,  from  a  practical  point  of  view  the  STURM  Image-Lines 
perpendicular  to  the  chief  ray  of  the  bundle  possess  also  certain  ad- 
vantages; by  their  very  definition,  they  have  the  distinguishing  prop- 
erty of  being  the  places  in  the  bundle  where  the  element  of  wave-sur- 
face is  smallest.  The  relative  merits  of  the  two  modes  of  defining  the 
image-lines  are  discussed  very  thoroughly  by  CZAPSKI  in  the  paper 
referred  to  (of  which  the  above  is  a  digest  and  partial  translation), 
and  the  conclusion  which  he  reaches  is  that  he  can  see  no  advantage 
in  abandoning  the  "classical"  image-lines  of  STURM.  See  also  §  232. 

1  See  L.  MATTHIESSEN:  Berlin- Eversbusch.  Zs.  /.  vergl.  Augenheilk.,  vi.  (1889),  p.   104. 


CHAPTER  III. 

REFLEXION  AND  REFRACTION  OF  LIGHT-RAYS  AT  A  PLANE  SURFACE. 
ART.  16.     THE  PLANE  MIRROR. 

50.  In  the  diagram  (Fig.  16)  the  plane  reflecting  surface  or  mirror 
is  supposed  to  be  perpendicular  to  the  plane  of  the  paper ;  the  straight 
line  AB  showing  its  trace  in  this  plane.  The  reflected  ray  BQ  cor- 


PATH  OF  A  RAY  REFLECTED  AT  A 
PLANE  MIRROR. 


HOMOCENTRIC      BUNDLE      OF      RAYS      RE- 
FLECTED AT  A  PLANE  MIRROR. 


responding  to  an  incident  ray  LB  will  lie  in  the  plane  of  incidence 
(which  is  here  the  plane  of  the  paper),  and  the  path  of  the  reflected 
ray  will  be  such  that,  if  it  is  produced  backwards  to  meet  at  Z/,  the 
straight  line  drawn  from  L  perpendicular  to  the  plane  of  the  mirror 
at  the  point  A,  we  shall  have  L' A  =  AL  (see  §  28).  Moreover,  since 
the  position  of  the  point  L'  is  independent  of  the  position  on  the  plane 
mirror  of  the  incidence-point  B,  all  incident  rays  which  go  through 
the  point  L  will  be  reflected  along  paths  which,  prolonged  backwards, 
will  meet  in  the  point  Lf  (Fig.  17);  so  that  to  a  homocentric  bundle  of 
incident  rays  reflected  at  a  plane  mirror  there  corresponds  also  a  homo- 
centric  bundle  of  reflected  rays. 

The  points  designated  by  L  and  Z/,  which  are  the  vertices  of  these 

51 


52  Geometrical  Optics,  Chapter  III.  [  §  50. 

two  corresponding  homocentric  bundles  of  rays,  are  a  pair  of  conju- 
gate points  with  respect  to  the  plane  mirror  (§44).  In  this  case  the 
point  L'  is  a  virtual  image  of  the  object-point  L.  But  if  the  bundle 
of  incidence  rays  converge  towards  a  'Virtual  object-point"  L  situated 
behind  the  mirror,  we  shall  have  then  a  real  image  at  a  point  L'  in 
front  of  the  mirror.  Thus,  the  object-point  L  may  be  situated  any- 
where in  infinite  space,  and  there  will  be  always  a  corresponding  image- 
point  L'.  It  may  be  remarked  here  that  the  plane  mirror  is  the  only 
optical  system  which,  without  any  restrictions  whatever  as  to  the 
angular  apertures  of  the  bundles  of  rays  concerned  in  the  formation 
of  the  image,  satisfies  perfectly  the  geometrical  condition  of  collinear 
correspondence,  viz.,  that  to  every  object-point  there  shall  correspond 
one,  and  only  one,  image-point. 

The  straight  line  LL'  is  bisected  at  right  angles  by  the  plane  of 
the  mirror;  and  hence  if  we  put 

AL  =  v,     AL'  =  v', 

that  is,  if  the  symbols  v,  vf  denote  the  abscissae,1  with  respect  to  the 
point  A  as  origin,  of  the  points  L,  L',  respectively,  we  may  write  the 
so-called  abscissa-equation  for  the  case  of  reflexion  at  a  plane  mirror, 
as  follows: 

ds) 


whereby,  knowing  the  position  of  the  object-point  L,  we  can  ascertain 
the  position  of  the  corresponding  image-point  L'. 

1  The  word  abscissa  will  be  employed  throughout  this  book  (unless  otherwise  specific- 
ally stated)  to  describe  the  position  of  the  point  where  a  ray  crosses  the  optical  axis  xx' 
of  a  refracting  or  reflecting  surface  with  respect  to  the  vertex  A  of  the  surface  as  origin. 
This  optical  axis  (which  will  be  particularly  defined  in  a  following  chapter)  is  identical 
here  with  the  straight  line  drawn  from  the  luminous  point  perpendicular  to  the  surface. 
Thus,  for  example,  in  Fig.  16,  the  abscissa  of  the  object-point  L  is  AL,  which  is  always  to 
be  reckoned  in  the  sense  in  which  the  letters  are  written,  so  that  if  v  =  AL,  then  —  v  —  LA. 

So  far  as  our  immediate  purposes  in  this  chapter  are  concerned,  it  is  entirely  imma- 
terial which  direction  along  the  axis  we  take  as  the  positive  direction;  the  opposite  direct- 
ion will,  of  course,  have  to  be  reckoned  as  negative.  Subsequently,  we  shall  see  that, 
as  a  rule,  it  will  be  convenient  to  reckon  the  positive  direction  along  any  ray  of  light  as  the 
direction  which  the  light  pursues  along  that  ray;  and  we  may,  therefore,  use  this  method 
here  (cf.  §  26).  Generally,  in  all  our  diagrams  the  incident  light  will  be  represented  as 
travelling  from  left  to  right. 

In  this  place  we  take  occasion,  also,  to  say  expressly  that,  if  A,  B,  C,  D  .  .  .  J,  K 
designate  the  positions  of  a  number  of  points  ranged  all  along  a  straight  line,  in  any  order 
whatever,  we  have  always  the  following  relations: 

AB  +  BA  =  o,     or     AB  —  —  BA; 

AB  +  BC  +  CA=o; 

AB  -f  BC  -f  CD  -f   •  •  •  +  JK  —  A  K-,  etc.,  etc.     (See  Appendix.) 


§50.] 


Reflexion  and  Refraction  of  Light-Rays. 


53 


. 


fci 


FIG.  18. 

IMAGK  OF  EXTENDED  OBJECT  IN  A  PLANE 
MIRROR. 


If,  instead  of  a  single  luminous  point  L,  we  have  an  extended  object 
consisting  of  an  aggregation  or  system  of  luminous  points,  to  each 
point  of  the  object  there  will  correspond  one  image-point,  and  the 
image  of  such  an  object  will  be 
formed  by  the  system  of  image- 
points.      Thus,    if    Llt    L2,   etc. 
(Fig.  1 8) ,  designate  the  positions 
of  the  points  of  the  object,  the 
positions  of    the   corresponding 
image-points    L[,  L'2,   etc.,  will 
be  determined  by  the  fact  that 
the  plane  of  the  mirror  must  bi- 
sect at  right  angles  the  straight 
lines  joining  each  pair  of  conju- 
gate points.     It  is  obvious  that 

the  image  in  this  case  will  have  exactly  the  same  dimensions  as  the 
object.  In  general,  however,  the  two  will  not  be  congruent;  that  is, 
image  and  object,  although  similar  and  equal,  cannot  be  superposed, 
because,  being  symmetrically  situated  with  respect  to  the  mirror, 
their  corresponding  parts  face  opposite  ways;  so  that,  for  example, 
the  situation  of  the  object  with  respect  to  right  and  left  is  reversed 

in  the  image.  The  object  and  image  will  be 
congruent  only  in  case  the  object  is  a  plane 
figure,  as  shown  in  the  diagram. 

The  extent  of  the  portion  of  the  mirror  that 
is  actually  utilized  will  depend  on  the  magni- 
tude and  position  of  the  object  whose  image 
is  to  be  viewed  and  also  on  the  position  of  the 
eye  of  the  observer.  Thus,  for  example,  in 
order  that  a  man  standing  erect  in  front  of  a 
vertical  plane  mirror  may  be  able  to  view  his 
image  from  head  to  foot,  the  height  of  the 
mirror  must  be  at  least  half  the  height  of  the 
man,  and  then  the  lower  edge  of  the  mirror 
must  be  placed  at  a  level  half-way  between 
the  levels  of  the  eyes  and  feet ;  as  may  easily 
be  verified. 

Wherever  the  eye  is  placed  in  front  of  a  plane  mirror,  the  image 
of  an  object  will  appear  always  at  the  same  place  and  of  the  same 
dimensions.  The  more  inclined  towards  the  mirror  is  the  cone  of  rays 
that  enter  the  eye,  the  greater  will  be  the  piece  of  the  mirror  utilized 


The  two  bundles  of  reflected 
rays  having  equal  angular 
apertures  intercept  unequal 
pieces  of  the  plane  mirror. 


54  Geometrical  Optics,  Chapter  III.  [  §  51. 

by  these  rays.  For  example  (Fig.  19),  the  cone  of  rays  which  enters 
the  eye  at  El  intercepts  on  the  mirror  a  shorter  piece  of  the  mirror 
than  will  be  intercepted  when  the  eye  is  placed  at  E2  in  the  figure. 
The  nearer  the  object  and  the  eye  are  to  the  mirror,  and  the  farther 
they  are  from  one  another,  the  greater  will  be  the  piece  of  the  mirror 
that  will  be  utilized  in  viewing  the  image.  If  the  surface  of  the  mirror 
is  not  accurately  plane,  any  irregularities  in  it  will  be  made  apparent 
by  viewing  the  image  at  very  oblique  incidence;  for  in  this  case  each 
element  of  the  mirror  that  is  used  will  produce  an  image,  and  the 
resulting  image  will  be  more  or  less  blurred  or  indistinct.  In  this 
way  it  is  possible  to  test  with  a  high  degree  of  accuracy  whether  a  sur- 
face is  truly  plane  or  not.  The  method  has  been  employed  to  show 
the  curvature,  due  to  the  spherical  form  of  the  earth's  surface,  of  the 
free  surface  of  tranquil  mercury. 

51.  A  number  of  the  most  important  uses  of  the  plane  mirror  de- 
pend on  the  fact  that  when  the  mirror  is  turned  through  any  angle  about 
an  axis  perpendicular  to  the  plane  of  incidence,  the  reflected  ray  will 
be  turned  through  an  angle  just  twice  as  great.  This  follows  imme- 
diately from  the  law  of  reflexion.  For  if  the  plane  of  the  mirror  is 
turned  through  any  angle  6,  the  normal  to  the  mirror  will  be  turned 
through  an  equal  angle,  and  hence  the  angle  between  a  given  incident 
ray  and  the  normal  at  the  point  of  incidence  will  be  changed  by  the 
amount  6,  and  therefore  the  angle  between  the  incident  and  the  re- 
flected rays  will  have  been  increased  (or  diminished)  by  26.  It  was 
POGGENDORFF  who  first  suggested  the  method  of  measuring  small 
angles  which  depends  on  this  principle,  and  which  has  been  extensively 
employed  for  this  purpose  in  a  great  variety  of  scientific  instruments, 
such  as  the  reflexion-lever,  the  mirror  galvanometer,  GAUSS'S  mag- 
netometer, etc.  Essentially  the  same  idea  is  employed  in  the  goni- 
ometer in  measuring  the  angles  of  crystals  and  prisms. 
:  In  this  connection,  we  may  mention  here  also  the  case  of  two  plane 
mirrors  at  which  the  rays  are  reflected  back  and  forth  alternately. 
The  incident  rays  emanating  from  a  luminous  point  placed  anywhere 
in  the  dihedral  angle  between  the  planes  of  the  two  mirrors  which 
fall  on  mirror  No.  I  will  give  rise  to  one  series  of  images,  while  the 
incident  rays  which  fall  on  mirror  No.  2  will  give  rise  to  a  second  series 
of  images.  The  images  of  both  series  will  evidently  all  be  ranged 
on  the  circumference  of  a  circle  whose  centre  is  at  the  point  of  the  line 
of  intersection  of  the  planes  of  the  mirrors  determined  by  a  plane 
through  the  luminous  point  perpendicular  to  this  line,  and  whose 
radius  is  equal  to  the  length  of  the  straight  line  joining  the  centre  with 


§  52.]  Reflexion  and  Refraction  of  Light-Rays.  55 

the  luminous  point.  The  last  image  of  each  of  the  two  series  will  be 
the  first  image  of  that  series  which  is  so  situated  as  to  be  behind  both 
mirrors,  and  which  lies,  therefore,  in  the  equal  dihedral  angle  formed 
by  continuing  the  planes  of  the  mirrors  backwards  beyond  their  com- 
mon line  of  intersection.  The  total  number  of  images  in  any  case 
will  depend  on  the  angle  included  between  the  planes  of  the  two 
mirrors,  and  also  on  the  position  of  the  object-point  with  respect  to 
the  mirrors.  If  6  denotes  the  angle  between  the  two  plane  mirrors, 
and  if  the  angular  distances  of  the  object-point  from  the  two  mirrors 
are  denoted  by  co  and  <p,  so  that  0  =  co  +  <p,  the  total  number  of 
images  may  be  shown  to  be  as  follows: 

(1)  If  the  angle  6  is  contained  a  whole  number  of  times,  say  i,  in 
1 80°,  so  that  i  So/ 6  =  i,  the  number  of  images  in  this  case  will   be 
21  —  i,  no  matter  what  may  be  the  values  of  the  angles  denoted  by 
co,  <p. 

(2)  But  if  the  angle  9  is  contained  in  180°  a  whole  number  of  times 
i  with  a  remainder  e  <  6,  so  that   180/6  =  i  +  e/0,  there   are   four 
cases  here  to  be  distinguished  as  follows: 

(a)  If  e  >  6/2,  the  number  of  images  in  this  case  =  2^  +  2; 

(b)  If  e  =  6/2,  the  number  of  images  in  this  case  =  2^+1; 

(c)  If  €  <  0/2,  but  >  co,  the  number  of  images  =  2*  +  i ; 
and 

(d)  If  e  <  (p  and  also  <  co,  the  number  of  images  =  2i. 

See  HEATH'S  Geometrical  Optics  (Cambridge,  1887),  Art.  32. 

This  theory  explains  Sir  DAVID  BREWSTER'S  Kaleidoscope,  in  which 
multiple  images  are  formed  by  two  plane  mirrors  inclined  to  each 
other.  When  the  mirrors  are  parallel  and  facing  each  other  (6  =  o), 
the  number  of  images  will  be  infinite. 

Another  theorem  of  inclined  mirrors  given  in  HEATH'S  Geometrical 
Optics,  Art.  14,  which  is  applied  in  the  instrument  known  as  the  Sextant, 
is  as  follows: 

When  a  ray  of  light  is  reflected  an  even  number  of  times  (21)  in 
succession  at  two  plane  mirrors  (the  reflexions  occurring  in  a  plane 
at  right  angles  to  the  planes  of  the  mirrors),  the  total  deviation  is 
equal  to  2i  times  the  angle  of  inclination  of  the  mirrors. 

ART.  17.  TRIGONOMETRIC  FORMULAS  FOR  CALCULATING  THE  PATH  OF  A 

RAY  REFRACTED  AT  A  PLANE  SURFACE.  IMAGERY  IN  THE  CASE  OF 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  PLANE  SURFACE. 

52.  Let  L  (Fig.  20)  designate  the  position  of  a  point  on  a  ray 
incident  at  the  point  B  on  a  plane  refracting  surface  which  separates 


56 


Geometrical  Optics,  Chapter  III. 


[§52. 


two  isotropic  optical  media  of  absolute  indices  of  refraction  n  and  nf . 
The  straight  line  LA  or  x  which,  going  through  the  point  L,  meets 
the  surface  normally  at  the  point  A  is  called  the  axis  of  the  refracting 

plane  with  respect  to  the  point  L.     The 

magnitudes 

=  w.      Z  ALB  =  a. 


L     L 


M 

FIG.  20. 

RAY  OF  IJGHT    REFRACTED  AT  A 
PLANE. 

AL  =  v,    AL'  =  v'. 


which  determine  completely  the  position 
of  the  incident  ray,  may  be  called  the 
ray-co-ordinates.  Similarly,  if  L'  desig- 
nates the  position  of  the  point  where  the 
refracted  ray,  produced  backwards  from 
the  incidence-point  B,  crosses  the  axis 
x,  then 


will  be  the  ray-co-ordinates  of  the  refracted  ray  L'B.    The  problem 
is,  being  given  the  incident  ray  (v,  a),  to  determine  the  refracted  ray 

(*>',  <*')• 

From  the  figure  we  derive  immediately: 

v'  _  tan  a 
v       tan  a.' 

moreover,  by  the  law  of  refraction: 

n  •  sin  a  =  n'  •  sin  a' ; 
whence  are  obtained  the  following  formulae: 

1/-../2  2       •     2 

v  v  n    —  n  -sin  a 


n 


cos  a 


sin  a   =  —f  sin  a. 
n 


(16) 


These  equations  enable  us  to  find  the  magnitudes  v',  a'  and  to  deter- 
mine, therefore,  the  refracted  ray. 

For  a  given  value  of  v,  we  see  that  the  value  of  v'  will  depend  on 
the  angle  of  incidence  a.  Only  those  rays  emanating  from  the  point 
L  which  meet  the  plane  refracting  surface  at  equal  angles  of  incidence 
(and  which  lie,  therefore,  on  the  surface  of  a  right  circular  cone  gen- 
erated by  the  revolution  of  the  straight  line  LB  about  LA  as  axis) 


§53.] 


Reflexion  and  Refraction  of  Light-Rays. 


57 


will,  after  refraction,  all  intersect  at  a  point  L'  on  the  axis  x.  So 
that  if  L  is  a  luminous  point  emitting  rays  in  all  directions,  an  eye 
placed  in  the  second  medium  (nf)  will,  in  general,  not  see  a  distinct, 
but  only  a  blurred  and  distorted,  image  of  the  object-point  at  L;  as 
will  be  more  fully  explained  in  the  section  which  treats  of  the  caustic 
by  refraction  at  a  plane  surface  (Art.  18). 

53.  Refraction  of  Paraxial  Rays  at  a  Plane  Surface.  In  one 
special  case,  however,  the  imagery  produced  by  refraction  at  a  plane 
surface  is  ideal.  Let  MA  (Fig. 
21  )  be  the  axis  of  the  plane  re- 
fracting surface  /*  with  respect 
to  the  object-point  M  ;  and  let 
us  suppose  that  all  the  points  of 
the  refracting  plane  are  screened 
from  M  except  those  points 
which  are  infinitely  near  to  the 
point  A  where  the  axis  meets 
the  surface;  so  that  of  all  the 
rays  proceeding  from  M  only 
those  whose  paths  lie  very  close 
to  the  axis  can  meet  the  refract- 
ing surface.  We  shall  have  thus 
an  infinitely  narrow  bundle  of 
pamxial  incident  rays  (enor- 
mously exaggerated  in  the  dia- 
gram) whose  chief  ray  coincid- 
ing with  the  axis  of  the  refracting 
plane  meets  this  plane  normally. 
The  angles  of  incidence  and  re- 
fraction of  the  chief  ray  are  both 
equal  to  zero;  whereas  in  the 
case  of  all  the  other  rays  these 
angles  will  both  be  infinitely 
small.  If  we  suppose  that  the 
angles  a,  a'  are  so  small  that  we 
may  neglect  the  second  and 
higher  powers  of  these  angles, 
the  angle  a  disappears  entirely  from  the  first  of  equations  (16)  ;  and  if 
the  abscissae  with  respect  to  the  point  A  of  the  conjugate  axial  points 
M,  M'  are  denoted  by  u,  u',  respectively,  that  is,  if  here  we  put 


FIG.  21. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  PLANE. 
In  these  diagrams  the  incident  rays  are  supposed 
to  meet  the  Refracting  Plane  almost  normally. 
The  angular  apertures  of  the  cones  of  rays  are  in 
reality  infinitely  small,  although  they  are  here 
enormously  magnified.  Paraxial  Rays  diverging 
from  a  point  M  are  refracted  at  the  Plane  Surface 
as  though  they  came  from  M'. 

AM=u,    AM'  =  u', 


AM 


AM'  =  u', 


58 


Geometrical  Optics,  Chapter  III. 


§53. 


we  have  evidently  the  following  relation: 


(17) 


which  is  the  so-called  abscissa-equation  for  the  refraction  of  paraxial 
rays  at  a  plane  surface.  Provided  we  know  the  position  on  the  axis 
of  the  object-point  M,  this  equation  enables  us  to  determine  the  posi- 
tion of  the  corresponding  image-point  M'.  Thus,  to  a  homocentric 
bundle  of  incident  paraxial  rays  refracted  at  a  plane  surface  there 
corresponds  also  a  homocentric  bundle  of  refracted  rays. 

Within  the  infinitely  narrow  cylindrical  region  immediately  around 
the  axis  of  the  refracting  plane,  we  have,  therefore,  a  point-to-point 

correspondence  of  object  and  image. 
According  to  (17),  since  u,  u'  have 
the  same  signs,  the  points  M  ,  M  '  lie 
always  on  the  same  side  of  the  re- 
fracting  plane,  that  is,  the  point  Mf 
is  a  virtual  image  of  the  point  M. 
If  the  object  is  an  infinitely  short 
line  MQ  (Fig.  22)  perpendicular  to 
the  axis  at  M,  obviously,  the  image 
of  the  point  Q  will  be  a  point  Qr 
lying  on  the  straight  line  drawn 
through  Q  perpendicular  to  the  re- 
fracting plane  and  at  the  same  dis- 
tance from  this  plane  as  the  axial 
image-point  M'.  Consequently,  the 
image  of  the  infinitely  short  object-line  M  Q  at  right  angles  to  the  axis 
is  an  equal  and  parallel  line  M'Qf.  The  ratio  y'/y,  where  M  Q  =  y, 
M'Q'  =  y'  is  called  the  Lateral  Magnification  or  the  Linear  Magnifi- 
cation, and  will  be  denoted  here  by  the  symbol  Y.  Thus,  in  the  case 
of  the  imagery  produced  by  the  refraction  of  paraxial  rays  at  a  plane 
surface,  we  have: 


A/ 


M 


FIG.  22. 

IMAGERY  IN  THE  CASE  OF  REFRACTION 
OF  PARAXIAL  RAYS  AT  A  PLANE.  The  im- 
age of  the  infinitely  small  object-line  MQ 
parallel  to  the  Refracting  Plane  is  an  equal 
image-line  M'Q'  having  the  same  direction 
2&MQ. 

AM=u 


Y  =       = 


(18) 


The  two  equations  (17)  and  (18)  show  that  the  image  is  always  vir- 
tual and  erect  and  of  the  same  size  as  the  object,  provided  the  latter 
is  a  line  at  right  angles  to  the  axis.  If  £AMQ  is  not  a  right  angle, 
the  image-line  will  not  be  parallel  to  the  object-line  nor  of  the  same 
length  as  the  object-line.  We  have  here,  in  fact,  a  special  case  of 


§54.] 


Reflexion  and  Refraction  of  Light-Rays. 


59 


collinear  correspondence,  known  as  Central  Collineation,  the  refracting 
plane  being  itself  the  plane  of  collineation  and  the  centre  of  collinea- 
tion  being  the  infinitely  distant  point  of  a  straight  line  perpendicular 
to  the  refracting  plane.  It  is  the  relation  that  in  geometry  is  called 

affinity. 

ART.   18.     CAUSTIC  SURFACE  IN  THE   CASE   OF  A  HOMOCENTRIC  BUNDLE 
OF  RAYS  REFRACTED  AT  A  PLANE  SURFACE. 

54.  In  general,  as  we  saw  (§  52),  to  a  homocentric  bundle  of  rays 
incident  on  a  plane  refracting  surface  there  corresponds  a  system  of 
refracted  rays  which  is  not  homocentric.  It  will  be  an  instructive 
exercise  to  investigate  in  this  comparatively  simple  case  the  form  of 
the  caustic  surface  (§  46),  especially  as  this  example  will  afford  a  very 
good  illustration  of  the 
general  principles  ex- 
plained in  Art.  15  of  the 
preceding  chapter. 

Let  the  vertex  of  the 
homocentric  bundle  of 
incident  rays  be  desig- 
nated by  S  (Fig.  23),  and 
let  the  straight  line 
marked  /z  show  the  trace 
in  the  plane  of  the  paper 
of  the  refracting  plane. 
Since  everything  is  sym- 
metrical with  respect  to 
the  normal  5^4,  drawn 
from  S  to  the  plane  re- 
fracting surface,  it  will 
be  sufficient  to  investi- 
gate the  form  of  the  re- 
fracted wave-surface  in 
the  plane  of  the  paper. 
Let  the  straight  line  SB 
drawn  in  the  plane  of  the 
paper  and  meeting  the 
refracting  plane  in  the  point  B  represent  the  path  of  an  incident  ray, 
and  let  L'  designate  the  point  where  the  corresponding  refracted  ray, 
produced  backwards,  intersects  the  straight  line  SA .  In  the  case  which 
we  shall  consider  here  the  first  medium  (n)  is  supposed  to  be  optically 


t* 

FIG.  23. 

SPHERICAL  WAVE  DIVERGING  FROM  A  POINT  6"  AND  RE- 
FRACTED AT  A  PLANE  INTO  AN  OPTICALLY  RARER  MEDIUM 
(«'  <  n).  SB  is  ray  incident  on  refracting  plane  at  the 
point  B,  and  BPis  the  corresponding  refracted  ray. 


60  Geometrical  Optics,  Chapter  III.  [  §  54. 

denser  than  the  second  medium  (w')>  as,  for  example,  when  the  rays 
are  refracted  from  water  into  air;  hence,  n  >  nf,  where  n,  nf  denote 
the  absolute  indices  of  refraction  of  the  two  media.  In  this  case, 
therefore,  the  point  L'  will  lie  between  S  and  A,  as  shown  in  the  figure. 
Produce  the  normal  SA  into  the  second  medium  to  a  point  Q  so 
that  AQ  =  SA,  and  pass  a  circle  through  the  points  5,  B  and  Q,  and 
produce  the  refracted  ray  backwards  to  meet  the  circumference  of 
this  circle  in  the  point  designated  in  the  figure  by  K.  The  angle  SKQ 
is  evidently  bisected  by  the  straight  line  KB,  and  we  have: 

Z.SKB  =  Z.BKQ  =  ^BSQ  =  a, 

since  these  inscribed  angles  stand  on  equal  arcs  of  the  circle.  The 
two  angles  at  Lr  are  equal  to  the  angle  of  refraction  a'  and  to  the 
supplement  of  this  angle;  hence,  in  the  triangle  SL' K  we  have: 

SL' :  KS  =  sin  a. :  sin  a' ; 
and,  similarly,  in  the  triangle  QKL': 

L'Q:KQ  =  sino::sma'; 
so  that,  by  the  law  of  refraction: 

SL'iKS  =  L'Q:KQ  =  n':n, 
or 

(SL'  +  L'Q}:(KS  +  KQ)  =  »':»; 
that  is, 

KS  +  KQ  =  -,  SQ  =  constant. 

Thus,  we  see  that  the  locus  of  the  point  K  is  an  ellipse  with  its  foci 
at  the  points  5  and  Q.  Moreover,  the  refracted  ray,  which  bisects 
the  angle  SKQ,  is  normal  to  the  ellipse  at  K.  The  ellipse  is,  therefore, 
an  orthotomic  curve  for  the  system  of  refracted  rays  which  lie  in  the 
plane  of  the  paper. 

The  meridian  section  of  the  refracted  wave-front  at  any  moment 
may  be  found  by  measuring  off  equal  distances  from  the  points  of 
this  ellipse  along  each  refracted  ray;  that  is,  the  refracted  wave-fronts 
are  parallel  curves  to  this  orthotomic  ellipse  (Fig.  24).  These  curves 
will  not  be  themselves  ellipses,  since  the  parallel  to  a  conic  is,  in  gen- 
eral, a  curve  of  the  eighth  degree.1  But  the  parallel  to  a  conic  has  the 
same  evolute  as  the  conic  itself;  so  that  the  caustic  curve  which  is  the 

1  See  SALMON'S  Conic  Sections,  6th  edition,  Art.  372,  Ex.  3. 


§  55.]  Reflexion  and  Refraction  of  Light-Rays.  61 

evolute  of  the  wave-line  will  in  the  present  case  be  the  evolute  of  an 
ellipse. 

If  here  we  put  AS  =  c  (Fig.  23),  and  if  the  centre  A  of  the  ellipse 
is  taken  as  origin  of  a  system  of  rectangular  axes  (SA,  AB  being  the 

Wave   Front 


L  urn  inous 
Point 

FIG.  24. 

CAUSTIC  CURVE  AND  WAVE-I^INE  IN  THE  CASE  OF  REFRACTION  OF  CIRCULAR  WAVES  AT  A 
STRAIGHT  lyiNE.    Rays  refracted  from  water  into  air. 

directions  of  the  positive  axes  of  x,  y,  respectively) ,  the  Cartesian  equa- 
tion of  the  ellipse  will  be: 


and  the  rationalized  equation  of  the  evolute  of  this  ellipse  is: 

{«V  +  (n2  -  n'*)y2  -  n'*c2}3  +  2jc2n2ri\n2  -  n'*)x2y2  =  o. 

This  is,  therefore,  the  equation  of  the  caustic  curve  in  the  case  here 
considered.  The  caustic  here  is  a  "virtual"  caustic. 

It  has  been  assumed  above  that  the  first  medium  was  optically  den- 
ser than  the  second.  In  the  opposite  case,  viz.,  n  <  ri,  the  ortho- 
tomic  curve  for  the  system  of  meridian  refracted  rays  proves  to  be  a 
hyperbola  with  the  same  foci  as  the  ellipse  above,  so  that  the  caustic 
curve  for  this  case  will  be  the  evolute  of  the  hyperbola. 

55.  The  equation  of  the  caustic  by  refraction  at  a  straight  line 
may  also  be  deduced  directly,  as  follows: 


62 


Geometrical  Optics,  Chapter  III. 


§55. 


Taking,  as  above,  the  point  A  (Fig.  25),  which  is  the  foot  of  the 
perpendicular  let  fall  from  the  object-point  S  on  the  refracting  straight 
^  line,  as  the  origin  of  a  system  of  rectangular 

axes,  where  SA  and  AB  are  the  positive 
directions  of  the  axes  x  and  y,  respectively, 
and  putting: 


H 


we  obtain  immediately  the  following  relations : 

j~ 
AB  =  -c-tan  a;     BG  =  -  c 


GB  •  cos  a'      cos  a'  •  da 

Bo     —  —- =   - =*--  C' 


da! 


cos2  a  -  da' 


Since 
we  have: 


FIG.  25. 

USED  IN  DERIVING  EQUA- 
TION OF  CAUSTIC  BY  REFRAC- 
TION AT  A  STRAIGHT  lyiNE. 
The  plane  of  the  paper  is  the 
plane  of  incidence  of  the  inci- 
dent ray  SB,  to  which  corre- 
sponds the  refracted  ray  whose 
direction  is  along  the  straight 
line  S'B.  G  is  a  point  in  the 
plane  of  incidence  on  the  re- 
fracting straight  line  and  in-  Hence,  eliminating  a',  we  obtain: 

finitely  near  to  the  incidence- 
point^.   SG.S'G  incident  and  '    »2 
refracted  rays. 


cos2  a ' 


sin  a  =  n  -sin  a. 


_  ,      n   cos  a 

da   =  — r ,  da. 

n   cos  a 


/    .2  o      •    2     \ 

c  (n    —  n  •  sin  a) 


nn'  cos3  a 


Now 


HS'  =  BS'-cosa' 
c 


nn 


Again,  since 

AB=-C'tana,    BH  =  BS'-sina,     and     AH 
we  find  after  several  reductions: 


AB  +  BH, 


AH 


tan3  a. 


Eliminating  tan  a  from  these  expressions  for  x  and  y,  we  obtain  the 
Cartesian  equation  of  the  locus  of  the  primary  image-point  S'  (§  47) 
corresponding  to  the  object-point  5,  as  follows: 


i; 


§55.] 


Reflexion  and  Refraction  of  Light-Rays. 


63 


which,  being  rationalized,  gives  precisely  the  same  equation  as  is  given 
at  the  end  of  §  54. 

The  caustic  turns  its  convex  side  towards  the  refracting  straight 
line  and  touches  it  at  a  point  designated  by  V  in  Fig.  26  whose  dis- 
tance from  the  point  A=  n'cjVn'  —  n2,  and  which  is  therefore  the 
extreme  point  of  incidence  on  the  positive  side  of  the  y-axis.  Of 
course,  there  is  also  another  point  of  tangency  at  an  equal  distance 
from  A  on  the  negative  side  of  the  3>-axis.  Thus,  for  example,  if  the 
radiant  point  S  is  in  water,  and  if  the  rays  emerge  from  water  into  air 
(njri  =  4/3),  we  shall  find  AV  =  1.14-  AS.  Putting  y  =  o  in  the 
equation  of  the  caustic  curve,  we  find  the  cusp  of  the  caustic  at 
a  point  M'  on  the  normal  to  the  refracting  surface  at  A,  such  that 
AM'  =  n'-ASfn,  and  hence  (§53)  this  point  M'  is  the  image-point 
by  paraxial  rays  of  the  object-point  S. 

If  the  diagram  is  revolved  around  SA  as  axis,  the  caustic  curve 
will  generate  the  caustic  surface  of  the 
refracted  rays  corresponding  to  the 
homocentric  bundle  of  incident  rays 
emanating  in  all  directions  from  the 
radiant  point  S.  There  are  always 
two  caustic  surfaces,  but  in  case  the 
refracting  surface  is  a  surface  of  revolu- 
tion, one  of  the  caustic  surfaces  col- 
lapses into  a  piece  of  the  axis  of  sym- 
metry, which  in  this  case  is  the  segment 
SM'  (see  §46). 

If  an  eye  were  placed  at  the  point 
E  in  air  (Fig.  26),  and  if  at  a  point  5 
below  the  surface  of  still  water  there 
were  situated  a  radiant  point,  the  pri- 
mary image  of  the  radiant  point  6"  of  the  still  water  is  the  refracting  plane, 

would  be  located  at  the  point  of  tan- 
gency S'  of  the  tangent  to  the  caustic 
curve  drawn  from  the  point  E.  If 
the  eye  is  placed  vertically  above  the 
radiant  point  5,  the  image  will  be  seen 
at  M',  that  is,  at  a  depth  one-fourth 
nearer  to  the  surface  of  the  water 
than  the  object-point  actually  is.  We 
see  therefore  how  it  is  that  an  object  under  water  viewed  by  an 
eye  in  the  air  above  will,  in  general,  appear  not  only  to  be  raised 


FIG.  26. 

CAUSTIC  BY  REFRACTION  AT  A  PLANE 
SURFACE  FOR  CASE  WHEN  n  >  n'.  Dia- 
gram is  drawn  for  the  case  when  the 
first  medium  is  water  and  the  second 
medium  is  air.  The  horizontal  surface 


the  rays   being   refracted  from  below 
upwards. 

The  refracted  ray  BE,  corresponding 
to  the  incident  ray  SB,  when  produced 
backwards  is  tangent  to  the  caustic 
curve  at  the  point  marked  S1  and  meets 
the  normal  SA  at  the  point  marked  S'. 
These  points  S',  S'  are  the  positions  of 
the  I.  and  II.  Image-Points  of  the  astig- 
matic bundle  of  refracted  rays  that  enter 
the  eye  at  E. 


64 


Geometrical  Optics,  Chapter  III. 


[§56. 


towards  the  surface,  but  also  to  be  displaced  sideways  more  and  more 
towards  the  observer,  the  more  obliquely  he  regards  the  object.  Ob- 
viously, incident  rays  meeting  the  water-surface  at  points  beyond  the 
extreme  point  V  will  be  totally  reflected  (§  27). J 

ART.  19.  ASTIGMATIC  REFRACTION  OF  AN  INFINITELY  NARROW  BUNDLE  OF 
RAYS  AT  A  PLANE  SURFACE. 

56.  In  the  diagram  (Fig.  27)  the  refracting  plane  designated  by  ju 
is  supposed  to  be  perpendicular  to  the  plane  of  the  paper.  The  point 
S  in  the  first  medium  (n)  is  the  vertex  of  an  infinitely  narrow  homo- 
centric  bundle  of  incident  rays,  whose  chief  ray,  viz.,  the  ray  SB  or 


FIG.  27. 

ASTIGMATIC  BUNDLE  OF  REFRACTED  RAYS  DUE  TO  REFRACTION  AT  A  PLANE  OF  AN  INFINITELY 
NARROW  HOMOCENTRIC  BUNDLE  OF  INCIDENT  RAYS,  u,  u'  are  the  chief  incident  and  refracted 
rays.  6" is  the  Object-Point;  S'  is  I.  Image- Point;  _«?  is  II.  Image- Point. 

u,  meets  the  refracting  plane  at  the  incidence-point  B.  The  section 
of  this  bundle  of  rays  made  by  the  refracting  plane  will  be  a  small 
closed  curve  GJGJ-,  which  will  be  elliptical  in  case  the  cone  of  incident 

1  See  in  connection  with  this  section  L.  MATTHIESSEN:  Das  astigmatische  Bild  des 
horizontalen,  ebenen  Grundes  eines  Wasserbassins :  Ann.  der  Phys.  (1901),  347~352. 


§  58.]  Reflexion  and  Refraction  of  Light-Rays.  65 

rays  has  a  circular  cross-section.  The  chief  refracted  ray  u' ,  corre- 
sponding to  the  chief  incident  ray  w,  will,  if  produced  backwards  from 
B,  meet  in  the  point  designated  by  S'  the  straight  line  SA  which  is 
normal  to  the  refracting  plane  at  the  point  A.  In  general,  the  other 
rays  of  the  bundle  of  refracted  rays  will  not  intersect  the  chief  ray 
u' ',  but  will  pass  from  one  side  of  it  to  the  other  both  above  and  below 
the  plane  of  the  paper.  The  refracted  rays  will  constitute  an  astigmatic 
bundle  of  rays  (§  47),  on  whose  chief  ray  u'  the  two  image-points  will 
lie.  In  order  to  ascertain  the  positions  of  these  image-points,  we  must 
investigate  those  rays  of  the  astigmatic  bundle  which  meet  the  chief 
ray  u'. 

57.  The  Meridian  Rays.     The  three  points  5,  B  and  A  determine 
the  plane  of  incidence  of  the  chief  ray  u;  in  the  diagram  this  is  the 
plane  of  the  paper.     This  is  likewise  the  plane  of  incidence  of  all  the 
rays  of  the  bundle  which  meet  the  refracting  plane  at  points  lying 
along  the  diameter  GG  of  the  closed  curve  GJGJ.     The  rays,  there- 
fore, which  are  refracted  at  the  points  G,  G  will  necessarily  meet  the 
refracted  chief  ray  u'  -,  and,  since  we  assume  that  the  bundle  of  rays 
is  infinitely  narrow,  the  rays  refracted  at  the  points  G,  G  will  meet  the 
chief  refracted  ray  u'  in  one  and  the  same  point  S',  provided  we  neg- 
lect infinitesimal  magnitudes  of  the  second  order;  and  the  same  thing 
will  be  true  of  all  the  rays  refracted  at  points  lying  along  the  line- 
element  GG.     The  plane  of  incidence  of  the  chief  ray  u  which  contains 
this  pencil  of  rays  is  one  of  the  principal  planes  of  curvature  (§  46)  of 
the  refracted  wave-surface  at  the  incidence-point  £,  and  these  rays 
are  the  so-called  meridian  rays  of  the  bundle.     The  meridian  rays  of 
the  refracted  bundle  of  rays  all  intersect  at  the  I.  Image-Point  Sf. 
In  this  statement  it  is  assumed  that  we  neglect  magnitudes  of  the 
second  order  of  smallness,  and  hence  the  convergence  of  the  rays  at 
Sf  is  said  to  be  a  "convergence  of  the  first  order"  only. 

Moreover,  to  a  pencil  of  incident  rays  proceeding  from  the  radiant 
point  S  and  meeting  the  refracting  plane  at  points  lying  along  a  chord  ' 
of  the  curve  GJGJ  which  is  parallel  to  the  diameter  GG  there  corre- 
sponds a  pencil  of  refracted  rays  lying  in  the  same  plane  as  the  pencil 
of  incident  rays  (the  plane  determined  by  the  chord  and  the  radiant 
point)  whose  vertex  will  be  a  point  infinitely  close  to  the  point  S', 
above  or  below  it,  lying  in  the  I.  Image-Line  at  S'  which  is  perpen- 
dicular to  the  plane  of  incidence  of  the  chief  ray  u  (§47). 

58.  The  Sagittal  Rays.     Let  us  next  consider  the  rays  of  the  in- 
finitely narrow  bundle  which  meet  the  refracting  plane  at  points  lying 
along  a  diameter  //  of  the  curve  GJGJ  which  is  at  right  angles  to  the 

6 


66  Geometrical  Optics,  Chapter  III.  [  §  58. 

diameter  GG.  The  rays  of  this  pencil  which  are  incident  at  the  end- 
points  J,  /  of  the  diameter  JJ  are  symmetrical  with  respect  to  the 
normal  5^4,  so  that  after  refraction  they  will  intersect  the  chief 
refracted  ray  u'  in  the  point  3'  where  u'  meets  SA .  This  can  be  made 
clearer,  if  necessary,  by  imagining  that  the  right  triangle  SBA  is 
rotated  through  an  infinitely  small  angle  above  and  below  the  plane  of 
the  paper  around  SA  as  axis,  so  that  the  point  B  traces  the  line-ele- 
ment //,  and  the  chief  incident  ray  u  coincides  in  succession  with  all 
the  rays  of  the  pencil  SJ J.  It  is  obvious  that  all  the  rays  of  this 
pencil  will,  after  refraction,  intersect  the  chief  refracted  ray  u'  at  the 
II.  Image-Point  S'.  The  plane  ~S'  JJ  which  is  the  plane  of  this  pencil 
of  refracted  rays  and  which  is  perpendicular  to  the  plane  of  incidence 
of  the  chief  incident  ray  u  is  the  other  principal  plane  of  curvature  of 
the  refracted  wave-surface  at  the  incidence-point  B.  This  plane 
determines  the  sagittal  section  of  the  bundle  of  refracted  rays,  and  the 
pencil  of  rays  S'JJ  contains  the  sagittal  rays  after  refraction;  these 
refracted  rays  correspond  to  the  incident  sagittal  rays  belonging  to 
the  pencil  SJJ.  The  rays  of  the  sagittal  section  of  the  bundle  of 
refracted  rays  intersect  in  Sf  not  merely  approximately,  but  exactly, 
because  in  the  sagittal  section  there  is  symmetry  with  respect  to  the 
plane  of  incidence  of  the  chief  ray  u,  so  that  rays  from  the  radiant 
point  5  which  make  equal  angles  with  the  plane  SAB  on  opposite 
sides  of  this  plane  will,  after  refraction,  all  pass  through  5';  so  that 
at  the  II.  Image-Point  the  convergence  is  of  the  second  order. 

To  a  pencil  of  incident  rays  which  meet  the  refracting  plane  at 
points  lying  along  a  chord  of  the  closed  curve  GJGJ  which  is  parallel 
to  the  diameter  //  there  corresponds  a  pencil  of  refracted  rays  which 
meet  all  at  one  point  of  the  II.  Image-Line.  This  latter  lies  in  the 
plane  of  incidence  of  the  chief  ray  u,  and,  according  to  STURM,  is  per- 
pendicular at  5'  to  the  chief  refracted  ray  u' .  However,  we  may  also 
consider  as  II.  Image-Line  of  the  astigmatic  bundle  of  refracted  rays, 
not  the  line-element  in  the  plane  of  incidence  of  the  chief  ray  u  that  is 
perpendicular  to  the  refracted  chief  ray  u'  at  the  point  S',  but  the  ele- 
ment mn  of  the  normal  to  the  refracting  plane  at  the  point  A  which  is 
intercepted  on  this  normal  by  the  two  extreme  rays  of  the  meridian 
pencil  of  refracted  rays.  Through  this  bit  of  the  normal  all  the  rays 
of  the  bundle  of  refracted  rays  must  pass,  as  may  easily  be  seen  by 
rotating  the  plane  of  the  paper  around  SA  as  axis  through  a  small 
angle  above  and  below  this  plane.  In  the  course  of  this  rotation,  the 
rays  of  the  meridian  section  will  trace  out  all  the  other  rays  of  the 
bundle,  but  the  element  mn  of  the  normal  5^4  will  remain  unchanged 
in  magnitude  and  in  position.  As  to  this  matter,  see  §  49. 


59.] 


Reflexion  and  Refraction  of  Light- Rays. 


67 


We  proceed  now  to  determine  the  positions  of  the  two  image-points 
Sf  and  Z>'  of  the  astigmatic  bundle  of  refracted  rays. 

59.  Position  of  the  Primary  Image-Point  5'.  In  the  diagram  (Fig. 
28)  the  straight  line  A  B  shows  the  trace  in  the  plane  of  the  paper  of  the 
refracting  plane.  The  straight 
line  SB  represents  the  chief 
ray  u  of  an  infinitely  narrow 
homocentric  bundle  of  incident 
rays  emanating  from  the  ob- 
ject-point S.  The  plane  of  the 
paper  is  the  plane  of  incidence 
of  the  chief  ray  u.  Infinitely 
near  to  the  incidence-point  B 
of  the  chief  ray  and  in  the 
plane  of  the  paper  let  us  take 
the  point  G,  so  that  SG  repre- 
sents  a  secondary  ray  of  the 
pencil  of  meridian  incident  rays. 


FIG.  28. 


REFRACTION  OF  NARROW  BUNDLE  OF  RAYS  AT 
A  PLANE.  Figure  for  the  determination  of  the  posi- 
tions of  the  I.  and  II.  Image-Points  S'  and  Sf  on  the 
chief  refracted  ray  u'  corresponding  to  the  object- 
point  .Son  the  chief  incident  ray  u. 


The  I.  Image-Point  5'  will  be 

at  the  point  of  intersection  of 

the  refracted  rays  correspond- 

ing  to   the   incident   rays  SB 

and  SG.     Let  a,  a'  denote  the  angles  of  incidence  and  refraction  of  the 

chief  ray;  so  that,  referring  to  the  figure,  we  may  write: 


NBS  = 


NBS'  =  a',      Z.  BSG  =  da,      /.  BS'G  =  da'. 


Then  in  the  triangles  BSGt  BS'G  we  have: 

BG         da          BG         da' 

and,  therefore: 


SB      cos  a'     S'B      cos  a/' 
BS'      cos  a'  da 


And  since 

we  obtain  finally: 

or  if  we  put  here: 


BS        cos  a  da'* 


n  •  sin  a  =  nf  •  sin  a', 


n'-cos2a' 


BS        n-cosza  ' 
BS  =  s,     BS'  =  s', 


68  Geometrical  Optics,  Chapter  III.  [  §  60. 

the  formula  above  may  be  written: 

,      n'.cos'q' 

5  '  =  -        —2  -  5.  (lo) 

n-cos  a 

If  the  chief  incident  ray  u  is  given  and  the  position  on  it  of  the 
radiant  point  5,  this  formula  enables  us  to  determine  the  position  of 
the  corresponding  I.  Image-Point  Sr  on  the  chief  refracted  ray  u'  . 

The  convergence-ratio  or  angular  magnification  of  the  rays  of  the 
meridian  section  is  the  ratio  da'  /da  of  the  angular  apertures  of  the 
pencils  of  incident  and  refracted  rays  in  the  meridian  section.  If  this 
ratio  is  denoted  by  the  symbol  Zu  (where  the  subscript  indicates  the 
chief  ray  of  the  pencil),  we  have  evidently: 


da       n  •  cos  a 


(20) 


60.  Position  of  the  Secondary  Image-Point  §'.  In  order  to  deter- 
mine the  position  of  the  II.  Image-Point  S',  which  is  at  the  point 
of  intersection  of  the  straight  line  drawn  through  the  homocentric 
object-point  5  perpendicular  to  the  refracting  plane  with  the  chief 
refracted  ray  u'  of  the  astigmatic  bundle  of  refracted  rays,  we  have 
from  the  triangle  SBS'\ 

BS_  _  sin  Z  B~$'S  _  sin  a' 
BS'  ~  sin  Z  BSS'  "  sin  a  ' 
and,  therefore: 

BS'  _  n' 
BS  ~  n' 
If  we  put 

BS  =  5,     BS'  =  s', 

we  shall  have  the  following  equation: 

-/     n'  /    \ 

S'=-s.  (2I) 

Thus,  if  we  know  the  position  of  the  homocentric  object-point  S  on  the 
chief  incident  ray  u,  this  formula  enables  us  to  locate  the  position  of 
the  II.  Image-Point  on  the  corresponding  chief  refracted  ray  u'. 

All  incident  rays  lying  on  the  surface  of  the  cone  generated  by  the 
revolution  of  the  ray  SB  around  the  normal  5^4  as  axis  will  after  re- 
fraction at  the  plane  refracting  surface  lie  on  the  surface  of  a  cone 
generated  by  the  revolution  of  ~S'  B  around  the  same  axis;  as  is  evident 
immediately  from  the  formula  just  obtained. 


§  62.]  Reflexion  and  Refraction  of  Light-Rays.  69 

If  SJ  is  a  ray  of  the  sagittal  section  of  the  homocentric  bundle  of 
incident  rays  which  meets  the  plane  refracting  surface  at  a  point  J 
infinitely  near  to  the  incidence-point  B  of  the  chief  incident  ray  u, 
S'J  will  show  the  direction  of  the  corresponding  ray  of  the  sagittal 
section  of  the  astigmatic  bundle  of  refracted  rays;  and  the  ratio  of 
the  angles  BS'J  and  BSJ  is  the  convergence-ratio  or  angular  magnifi- 
cation of  these  corresponding  pencils  of  sagittal  rays.  If  here  we  put: 

Z  BSJ  =  d\,     Z  BS'J  =  ffK', 

and  if  the  symbol  ZM  denotes  the  convergence-ratio  of  the  pencils  of 
incident  and  refracted  sagittal  rays  with  the  chief  incident  ray  u,  we 
have  evidently: 

_d\^  _  J$S^  _s_  _  n 

Z«~  d\  ~  BS'~s'~n'' 

61.  The  Astigmatic  Difference  of  the  bundle  of  refracted  rays  is 
the  piece  of  the  chief  refracted  ray  u'  comprised  between  the  II.  and 
I.  Image-Points  of  the  astigmatic  bundle  of  rays;  that  is, 

3'S'  =  S'B  +  BS'  =  s'  -  s'. 

In  the  case  of  an  infinitely  narrow  homocentric  bundle  of  incident 
rays  refracted  at  a  plane,  we  obtain  from  formulae  (19)  and  (21): 

sf       cos2  a 


and  for  the  astigmatic  difference  of  the  bundle  of  refracted  rays: 


_ 
n      cos  a 


The  astigmatic  difference  vanishes  only  in  case  a  =  a'  =  o;  that  is, 
when  the  chief  incident  ray  u  is  normal  to  the  refracting  plane  ;  which 
is  the  case  of  paraxial  rays  (§53). 

62.  Refraction  at  a  Plane  Surface  of  an  Infinitely  Narrow  Astig- 
matic Bundle  of  Incident  Rays.  If  the  bundle  of  incident  rays  is 
astigmatic,  and  if  we  designate  by  5  and  6"  (Fig.  29)  the  vertices  of  the 
pencils  of  incident  meridian  ana  /sagittal  rays,  respectively,  the  bun- 
dle of  refracted  rays  will,  in  general,  be  astigmatic  also,  and  the  I.  and 
II.  Image-Points  S'  and  S',  lying  on  the  chief  refracted  ray  u',  will 
correspond  to  the  points  5"  and  5,  respectively,  lying  on  the  chief  in- 
cident ray  u.  We  may  call  the  point  5  the  I.  Object-Point  and  the 


70 


Geometrical  Optics,  Chapter  III. 


[§63. 


point  3>  the  II.  Object- Point.  The  pencil  of  meridian  incident  rays 
emanating  from  the  I.  Object- Point  5  and  lying  in  the  plane  of  inci- 
dence of  the  chief  incident  ray  u  will  be  transformed  by  refraction  into 


FIG.  29. 

ASTIGMATIC  BUNDLE  OF  INCIDENT  RAYS  REFRACTED  AT  A  PLANE.  The  chief  rays  of  the  astig- 
matic bundles  of  incident  and  refracted  rays  are  designated  by  u  and  u'.  S,  S  designate  the  posi- 
tions on  u  of  the  I.  and  II.  Object- Points.  To  .Son  u  corresponds  the  I.  Image-Point  S'  on  u',  and 
to  .S  on  u  corresponds  the  II.  Image- Point  S'  on  u'.  In  the  diagram  the  plane  of  the  paper  coincides 
with  the  plane  of  the  meridian  rays. 

a  pencil  of  meridian  refracted  rays  lying  in  the  same  plane  with  its 
vertex  at  the  I.  Image-Point  S'.     Hence,  putting 

BS  =  s,     BS'  =  s', 
we  have,  according  to  formula  (19): 


n'  cos2  a' 


5'  =  — 


Similarly,  putting 

we  have  by  formula  (21) 


n  cos  a 
=  s,     B3'  •• 


—  s. 


The  bundle  of  incident  rays  will  have  been  rendered  astigmatic  in 
consequence,  for  example,  of  previous  refractions. 

ART.  20.     REFRACTION   OF  INFINITELY  NARROW  BUNDLE   OF  RAYS  AT  A 

PLANE:  GEOMETRICAL  RELATIONS  BETWEEN  OBJECT- 

POINTS  AND  IMAGE-POINTS. 

63.  If  on  a  given  incident  ray  u  (Fig.  30)  we  take  a  range  of  object- 
points  P,  Q,  R,  S,  •  -  -  ,  whereto  on  the  refracted  ray  u'  correspond  the 
range  of  I.  Image-Points  P',  Q1  ',  Rf,  S',  -  -  •  and  the  range  of  II.  Image- 
Points  P',  Q',  Tl',  3',  •  •  •  ,  then,  according  to  formula  (19),  we  must  have: 


BP' 
BP 


BQ' 
BQ 


BR' 
BR 


BS' 
BS 


which  means  that  the  straight  lines  PP',  QQ',  RR',  SS',  •  •  • ,  joining  the 


§  64.]  Reflexion  and  Refraction  of  Light-Rays.  71 

object-points  on  the  incident  ray  u  with  their  corresponding  I.  Image- 
Points  on  the  refracted  ray  u'  are  a  system  of  parallel  straight  lines; 
and,  hence,  the  point-ranges  P,  Q,  R,  •  •  •  and  P',  Qf,  R',  •  •_•  are  similar 
ranges  of  points.  And,  since  the  straight  lines  PP',  QQ',  RR',  •  -  - 


FIG.  30. 

REFRACTION  OF  NARROW  BUNDLE  OF  RAYS  AT  A  PLANE.  The  range  of  Object-Points  P,  Q,  ••• 
lying  on  the  chief  incident  ray  u  is  similar  to  the  range  of  I.  Image-Points  P',  Q' ,  •  •  •  and  also  to  the 
range  of  II.  Image- Points  P',  Q',  •  •  •  lying  on  the  chief  refracted  ray  u' . 

which  connect  the  Object- Points  on  the  incident  ray  u  with  their  cor- 
responding II.  Image-Points  on  the  refracted  ray  u'  are  all  perpendicu- 
lar to  the  refracting  plane,  and  therefore  parallel  to  each  other,  it 
follows  that  the  point-ranges  P,  Q,  R,  •  •  •  and  P',  ~Q',  R',  •  •  •  are  also 
similar  ranges  of  points. 

Conjugate  to  any  object-point  X,  lying  in  the  plane  of  incidence 
of  the  incident  ray  u,  there  will  be  on  the  refracted  ray  x',  which  cor- 
responds to  an  incident  ray  x  parallel  to  u  and  going  through  the 
object-point  X,  the  I.  Image-Point  X'  and  the  II.  Image-Point  X'\ 
and  the  range  of  Object-Points  lying  along  the  incident  ray  x  is  simi- 
lar to  the  ranges  of  I.  and  II.  Image-Points  lying  along  the  refracted 
ray  xf.  Thus,  the  plane  system  TJ  of  the  Object-Points  X,  •  •  • ,  which 
lie  in  the  plane  of  incidence  of  the  incident  ray  u,  is  in  affinity  with  the 
plane-system  r/  of  the  I.  Image-Points  X',  -  -  -  and  also  with  the  plane- 
system  ~r\'  of  the  II.  Image-Points  2"',  •  •  • :  hence,  also,  the  plane- 
systems  77 '  and  1?'  are  in  affinity  with  each  other.  The  straight  line  in 
which  the  plane  refracting  surface  meets  the  plane  of  incidence  is  the 
affinity-axis1  for  all  three  of  these  "affiri"  plane-systems. 

64.  Construction  of  the  I.  Image-Point.  Let  u  (Fig.  31)  be  an 
incident  ray  meeting  the  plane  refracting  surface  ju  at  the  point  B, 

1  The  affinity-axis  of  two  plane  "  affin  "  systems  is  the  straight  line  common  to  the 
two  systems  which  corresponds  with  itself  point  by  point.  Obviously,  any  pair  of  corre- 
sponding straight  lines  of  the  two  systems  will  meet  in  the  affinity-axis. 


72 


Geometrical  Optics,  Chapter  III. 


[§64. 


and  let  u'  be  the  corresponding  refracted  ray.  Corresponding  to  an 
Object-Point  5  on  u  we  find  the  II.  Image-Point  S'  on  u'  at  the  point 
of  intersection  with  u'  of  the  perpendicular  5^4  drawn  from  5  to  the 
refracting  plane.  Draw  SX,  Sf  Y  perpendicular  to  the  incidence- 
normal  B  N  at  X,  F,  respectively,  and  from  X  draw  XP  perpendicu- 
lar to  u  at  P,  and  from  F  draw  YPf  perpendicular  to  u'  at  P'.  Then 


and  since 


we  have: 


BP  =  BS-  cos2  a,     BP'  =  BS'  •  cos2  a'; 

1*5'=  ^ 
BP' 


w-cos  a 


Consequently,  according  to  formula  (19),  the  points  P,P'  are  corre- 
sponding points  of  the  "affin"  systems  17,  77';  so  that  if  P  is  an  Object- 
Point  of  the  chief  incident  ray  u,  P'  will  be  the  I.  Image- Point  lying 

u,  on  the  chief  refracted  ray 
u'.  The  I.  Image-Point 
S'  corresponding  to  the 
Object- Point  5  of  the 
chief  incident  ray  u  is 
found  by  drawing 
through  5  a  straight  line 
parallel  to  PP'  which, 
by  its  intersection  with 
the  chief  refracted  ray  u', 
will  determine  the  re- 
quired point  Sf.  This  is 
essentially  the  construc- 
tion given  by  REUSCH.1 
Another  construction 
of  the  two  correspond- 
ing points  of  the  "affin" 
systems  77,  t\'  is  as  follows : 
Through  5'  draw  a 
straight  line  perpendicular  to  the  refracted  ray  u'  and  meeting  the 
straight  line  BA  in  the  point  U]  draw  the  straight  line  US  meeting 
the  incidence-normal  B  N  in  the  point  Z;  and  from  Z  let  fall  a  per- 
pendicular ZQ  on  the  incident  ray  at  the  point  Q.  Then  the  point 
Sf  (or  Q')  is  the  I.  Image-Point  corresponding  to  the  Object-Point  Q 

1 E.  REUSCH:  Reflexion  und   Brechung  des  Lichts  an  sphaerischen  Flaechen  unter 
Voraussetzung  endlicher  Einf allswinkel :  POGG.   Ann.,  cxxx.    (1867),   497-517. 


FIG.  31. 

REFRACTION  OF  NARROW  BUNDLE  OF  RAYS  AT  A  PLANE. 
Construction  of  I.  Image-Point. 


§  64.]  Reflexion  and  Refraction  of  Light-  Rays.  73 

on  the  chief  incident  ray  u\  as  we  proceed  to  show.  Let  the  straight 
line  joining  U  and  S'  meet  the  incidence-normal  B  N  in  the  point  N; 
then 

B&__  BN_  BU_  BZL_BZL       BQ 

BP'  ~  AS'  ~  AU  ~  AS  ~  ~BX 
and,  therefore: 

BS' 


BQ  ~  BP 


and,  hence,  QS'  (or  QQf)  is  parallel  to  PP'.    Therefore,  S'  (or  Q') 
is  the  point  of  i)r  which  corresponds  to  the  point  Q  of  rj.1 

1  For  other  methods  of  construction  of  the  I.  Image-Point  see  F.  KESSLER:  Beitraege 
zur  graphischen  Dioptrik:  Zft.  f.  Math.  u.  Phys.,  xxix.  (1884),  65-74. 

See  also  the  construction  of  the  I.  Image  Point  in  the  case  of  refraction  at  a  plane 
considered  as  a  special  case  of  refraction  at  a  sphere,  as  given  in  §  249. 

Since  a  plane  surface  may  be  regarded  as  a  spherical  surface  with  its  centre  at  an  in- 
finite distance,  obviously,  all  the  problems  treated  in  this  chapter  can  be  considered  as 
special  cases  of  the  problem  of  refraction  at  a  spherical  surface,  as  will  be  seen  hereafter. 
According  to  this  view,  this  entire  chapter  might  be  regarded  as  superfluous. 


CHAPTER   IV. 

REFRACTION  THROUGH  A  PRISM  OR  PRISM-SYSTEM. 

ART.  21.     GEOMETRICAL    CONSTRUCTION    OF   THE    PATH    OF   A    RAY 

REFRACTED   THROUGH   A   PRISM   IN   A   PRINCIPAL 

SECTION    OF   THE   PRISM. 

65.  In  optics  the  term  Prism  is  applied  to  a  portion  of  a  trans- 
parent, isotropic  substance  included  between  two  non-parallel  plane 
refracting  surfaces  called  the  faces  or  sides  of  the  prism.  These  are 
distinguished  as  the  first  and  second  faces  of  the  prism  in  the  order 
in  which  the  light-rays  arrive  at  them.  The  straight  line  in  which 
the  two  plane  faces  meet  is  called  the  edge  of  the  prism,  and  the  dihe- 
dral angle  between  the  two  faces  is  called  the  refracting  angle.  This 
angle,  which  will  be  denoted  by  the  symbol  /3,  may  be  defined  more 
precisely  as  the  angle  through  which  the  first  face  of  the  prism  has  to  be 
turned,  around  the  prism-edge  as  axis,  in  order  to  bring  this  face  into 
coincidence  with  the  second  face.  A  principal  section  of  the  prism  is 
made  by  any  plane  perpendicular  to  the  edge  of  the  prism.  At  first 
we  shall  consider  only  such  rays  as  lie  in  a  principal  section  of  the 
prism  or  infinitely  narrow  bundles  of  rays  whose  chief  rays  lie  in  a  prin- 
cipal section. 

In  the  general  case  of  the  problem  of  refraction  through  a  single 
prism  we  have  to  do  with  as  many  as  three  optical  media,  viz.:  the 
medium  of  the  incident  rays  or  the  first  medium,  the  medium  of  which 
the  prism-substance  is  composed  and  the  medium  of  the  emergent  rays. 
The  absolute  indices  of  refraction  of  these  media  will  be  denoted  by 
nlt  n[  and  n'2  in  the  order  named.  In  most  cases  the  third  medium  is 
identical  with  the  first,  as,  for  example,  in  the  case  of  a  glass  prism 
surrounded  by  air;  and  unless  the  contrary  is  expressly  stated,  we 
shall  assume  that  this  is  the  case.  Thus,  we  shall  have  n^  =  n'2 ;  and 
the  symbol 


will  be  employed  to  denote  the  relative  index  of  refraction  of  the  me- 
dium of  the  prism-substance  with  respect  to  the  surrounding  medium. 
66.     The  following  construction  of  the  path  of  a  ray  refracted 

74 


§  66.]  Refraction  Through  a  Prism  or  Prism-System.  75 

through  a  prism  in  a  principal  section  was  published  by  REUSCH1  in 
1862;  the  same  construction  was  published  by  RADAU2  in  the  follow- 
ing year. 

In  the  diagrams  (Figs.  32  and  33)  the  plane  of  the  paper  represents 
a  principal  section  of  the  prism,  and  the  point   V  in  this  plane  shows 


FIG.  32. 


CONSTRUCTION  OF  THE  PATH  OF  A  RAY  REFRACTED  THROUGH  A  PRISM  IN  A  PRINCIPAL 
SECTION.    Case  when  nze  =  n\  and  n\'  >  n\. 

where  the  prism-edge  meets  the  plane  of  the  principal  section.  The 
two  plane  faces  of  the  prism,  designated  by  /*lf  /x2  are  shown  therefore 
by  two  straight  lines  meeting  in  the  point  V.  The  straight  line  LlBl 
(or  uj  represents  the  path  of  the  given  incident  ray  meeting  the  first 
face  of  the  prism  at  the  incidence-point  Bl ;  and  the  problem  is  to  con- 
struct the  remainder  of  the  ray-path  both  within  the  prism  and  after 
emergence  from  the  prism.  The  method  is  in  fact  the  same  as  that 
given  in  §29. 

With  the  point  V  as  centre,  and  with   radii   equal  to  r  and   r/n 

1  E.  REUSCH:  Die  Lehre  von  der  Brechung  und  Farbenzerstreuung  des  Lichts  an 
ebenen  Flaechen  und  in  Prismen  in  mehr  synthetischer  Form  dargestellt:  POGG.  Ann., 
cxvii.  (1862),  241-262. 

2R.  RADAU:  Bemerkungen  ueber  Prismen :  POGG.  Ann.,  cxviii.  (1863),  452-456.  The 
method  was  obtained  independently  by  RADAU  and  it  is  often  called  by  his  name;  but  he 
himself  in  CARLS  Rep.,  iv.  (1868),  p.  184,  acknowledges  REUSCH'S  priority  in  the  matter. 


76 


Geometrical  Optics,  Chapter  IV. 


[§66. 


(where  r  may  have  any  value),  describe  the  arcs  of  two  concentric 
circles.  Through  V  draw  a  straight  line  parallel  to  the  given  inci- 
dent ray  L^  meeting  the  circumference  of  circle  r/n  in  a  point  G, 
and  through  G  draw  a  straight  line  perpendicular  at  E  to  the  first 
face  of  the  prism  (produced,  if  necessary),  and  let  H  designate  the 


FIG.  33. 

CONSTRUCTION  OF  PATH  OF  RAY  REFRACTED  THROUGH  A  PRISM  IN  A  PRINCIPAL  SECTION. 
Case  when  «a'  =  n\  and  n\'  <  n\. 

position  of  the  point  where  this  straight  line  meets  the  circumference 
of  the  circle  r.  Then  the  straight  line  B1B2  drawn  parallel  to  the 
straight  line  VH  will  show  the  path  of  the  ray  within  the  prism.  For 
if  at  =  Z  N1B1L1  and  a[  =  Z.OB1B2  denote  the  angles  of  incidence 
and  refraction  at  the  first  face  of  the  prism,  then,  by  the  law  of  re- 
fraction : 

7-  sin  «   =  n-sm  a. 


According  to  the  construction,  we  have: 

sin  Z.EGV       VH 
sin  Z.EHV~  VG 


n 


and  since,  by  construction,  Z.EG  V=  alt  it  follows  that  Z  EHV  =  a'lt 

and  hence  the  path  of  the  ray  within  the  prism  must  be  parallel  to  VH. 

Again,  from  the  point  H  let  fall  a  perpendicular  on  the  second  face 


§67.] 


Refraction  Through  a  Prism  or  Prism-System. 


77 


of  the  prism,  meeting  this  face  in  the  point  designated  by  F  and  meet- 
ing the  circumference  of  the  circle  r/n  in  the  point  designated  by  /; 
then  the  straight  line  B2Q'2  drawn  parallel  to  the  straight  line  VJ  will 
represent  the  path  of  the  ray  after  refraction  at  the  second  face  of 
the  prism  back  into  the  first  medium.  For  if  a2  =  /.OB2Bl  and 
a'2  =  Z  N2B2Q'2  denote  the  angles  of  incidence  and  refraction  (emer- 
gence) at  the  second  face  of  the  prism,  we  must  have: 


We  have: 


sin  Z.FJV       VH 
sin  £FHV~  VJ 


n 


and,  since  by  construction  Z  FHV  =  a2,  it  follows  that  Z  FJV  =  a'2, 
and  hence  the  path  of  the  emergent  ray  is  parallel  to  VJ. 

In  Fig.  32  the  prism- 
medium  is  more  highly 
refracting  than  the  sur- 
rounding medium  (n  = 
ni/ni  greater  than  unity; 
as,  for  example,  a  glass 
prism  in  air).  The  op- 
posite case  (n  =  »l/#i 
less  than  unity,  as,  for 
example,  an  air  prism 
embedded  in  glass)  is 
shown  in  Fig.  33. 

If  the  medium  of  the 
emergent  rays  is  not  the 
same  as  that  of  the  inci- 
dent rays,  the  construc- 
tion is  practically  the 
same;  only,  around  Fas 
centre  we  must  describe 
now  the  arcs  of  three  con- 
centric circles  with  radii 
VH  =  r,  VG  =  »!?/»;, 
and  VJ  =  n2r/n[. 

67.  The  angle  GHJ  be- 
tween the  normals  to  the 

two  faces  of  the  prism  is  equal  to  the  refracting  angle  j8 ;  and  hence  for 
a  given  prism  this  angle  is  constant.     If  the  direction  of  the  incident 


y, 


FIG.  34. 

SHOWING  THE  PATHS  OF  THE  Two  LIMITING  RAYS  IN  A 
PRINCIPAL  SECTION  OF  A  PRISM,    (n  >  1.) 


78 


Geometrical  Optics,  Chapter  IV. 


68. 


ray  L1B1  is  varied,  the  vertex  H  of  this  angle  will  move  along  the  cir- 
cumference of  the  circle  of  radius  r,  the  sides  of  the  angle  having  the 
fixed  directions  of  the  normals  to  the  prism-faces.  The  two  extreme 
positions  of  this  point  H  which  are  reached  when  one  or  other  of  the 
sides  of  the  Z  GHJ  is  tangent  to  the  circle  of  radius  r/n  (which  can  occur 
only  when  n  is  greater  than  unity,  because  then  only  will  the  point  H 
lie  outside  the  circle  of  radius  r/n)  are  shown  in  Fig.  34.  The  two  inci- 
dent rays  which  correspond  to  these  two  extreme  positions  of  the  point 
H  are  the  ray  y^B^  which,  entering  the  first  face  of  the  prism  at 
'  'grazing"  incidence  (c^  =  90°)  at  the  point  Blt  and  traversing  the 
prism  as  shown  in  the  figure,  emerges  as  the  ray  y'2,  and  the  ray  zlt 
which,  entering  the  prism  at  the  point  Blt  and  arriving  at  the  second 
face  at  the  critical  angle  of  incidence  (§  27),  emerges  only  by  "grazing" 
this  face.  In  order  that  a  ray  incident  at  the  point  Bl  may  not  be 
totally  reflected  at  the  second  face  of  the  prism,  it  must  lie  within  the 


68.  When  the  point  H  (Fig.  32)  lying  on  the  circumference  of  the 
circle  of  radius  r  has  such  a  position  that  the  sides  of  the  Z  G  HJ  inter- 
cepted between  the  two 
concentric  circles  are  equal, 
that  is,HG=  HJ,  the  diag- 
onal VH  of  the  quadrilat- 
eral VGHJ  is  normal  to 
the  bisector  of  the  refract- 
ing angle  of  the  prism.  The 
special  positions  of  the 
points  G,  H  and  /  in  this 
case  may  be  designated  by 
GO,  #0,  and  J0  (Fig.  35). 
The  ray  which  traverses 
the  prism  parallel  to  the 
straight  line  VH0  is  sym- 
metrically situated  with  re- 
spect to  the  two  faces  of 
the  prism,  so  that  the  tri- 

angle VBlB2tQ  is  isosceles,  and  the  angles  of  incidence  at  the  first  face 
and  emergence  at  the  second  face  are  equal. 

The  angle  J  VG,  denoted  by  e,  between  the  directions  of  the  inci- 
dent and  emergent  rays  is  called  the  angle  of  deviation,  and  it  may  be 
shown  that  when  the  ray  traverses  the  prism  symmetrically,  this 
angle  has  its  least  value.  Let  H  (Fig.  35)  designate  the  position  of  a 


FIG.  35. 
PATH  OF  THE  RAY  OF  MINIMUM  DEVIATION. 


§  69.]  Refraction  Through  a  Prism  or  Prism-System.  79 

point  on  the  circumference  of  the  circle  of  radius  r  which  is  infinitely 
near  to  the  point  HQ,  and  draw  HG,  HJ  normal  to  the  two  faces  of  the 
prism  and  meeting  the  circumference  of  the  other  construction-circle 
in  the  points  G,  J,  respectively.  In  the  diagram  the  point  H  is  taken 
below  the  point  H0,  and  it  is  obvious  that  the  two  parallels  HQJQ,  HJ 
meet  the  circumference  of  the  circle  of  radius  r/n  more  obliquely  than 
do  the  parallels  H0GQ,  HG',  so  that  the  infinitely  small  arc  JJ0  is 
greater  than  the  infinitely  small  arc  GGQ',  and,  hence,  Z  JVJQ  is  greater 
than  /.GVGQ;  and,  consequently: 

Z.JVO  Z/0FG0. 

The  angle  JQVG0  is  the  angle  of  deviation  of  the  ray  which  traverses 
the  prism  symmetrically;  it  may  be  denoted  by  e0.  According  to  the 
above,  we  have  always  (for  we  shall  arrive  at  the  same  result  if  we 
take  the  point  H  on  the  other  side  of  HQ  from  that  shown  in  the  figure)  : 


Thus,  we  see  that  the  ray  which  traverses  the  prism  symmetrically  is 
also  the  ray  which  is  least  deviated  by  its  passage  through  the  prism  (Ray 
of  Minimum  Deviation).1 

69.  When  a  ray  of  light  passes  through  a  prism  the  material  of  which 
is  more  highly  refracting  than  the  surrounding  medium  (n  >  i),the  devia- 
tion is  always  away  from  the  edge  towards  the  thicker  part  of  the  prism. 
If  the  angles  of  the  triangle  VB1B2  (Fig.  32)  at  Bl  and  B2  are  both 
acute  angles,  the  incident  and  emergent  rays  lie  on  the  sides  of  the 
normals  at  Bl  and  B2  away  from  the  prism-edge,  so  that  at  each  re- 
fraction the  ray  will  be  bent  away  from  the  edge.  If  one  of  the  angles, 

1  On  the  subject  here  considered  optical  literature  is  very  extensive.  A  complete  list 
of  references  to  all  the  writers  will  be  found  in  H.  KAYSER'S  Handbuch  der  Spectroscopie, 
Bd.  I.  (Leipzig,  1900),  pages  258-^9.  Among  the  more  important  synthetic  proofs  of  the 
fact  that  the  ray  which  traverses  the  prism  symmetrically  is  the  least  deviated,  the  fol- 
lowing contributions  may  be  specially  mentioned: 

E.  LOMMEL:  Ueber  die  kleinste  Ablenkung  im  Prisma:  POGG.  Ann.,  clix.  (1876),  p.  329. 

F.  KESSLER:   Jahresbericht  der  Gewerbeschule  zu  Bochum  filr  1880.     Also,  Das  Mini- 
mum der  Ablenkung  eines  Lichtstrahls  durch  ein  Prisma:  WIED.  Ann.,  xv.  (1882),  333. 

R.  H.  SCHELLBACH:  Das  Minimum  der  Ablenkung  eines  Lichtstrahles  im  Prisma: 
WIED.  Ann.,  xiv.  (1881),  367. 

FR.  ,C.  G,  MUELLER:  Der  Satz  vom  Minimum  der  Ablenkung  beim  Prisma:  Zft.  f.  den 
phys.  u.  chem.  Unterr.,  iii.  (1889-90),  247. 

J.  H.  KIRKBY:  Refraction  through  a  prism:    Nature,  xliv.  (1891),  294. 

A.  KURZ:  Die  kleinste  Ablenkung  im  Prisma:  Zft.  f.  Math.  u.  Phys.,xxxvii.  (1892), 
317  and  xxxviii.  (1893),  319. 

H.  VEILLON:  Elementare  geometrische  Behandlung  des  Minimums  der  Ablenkung 
beim  Prisma:  Zft.  f.  den  phys.  u.  chem.  Unterr.,  xii.  (1899),  i5o-'2. 


80  Geometrical  Optics,  Chapter  IV.  [  §  70. 

say,  the  angle  at  B2,  is  a  right  angle,  there  will  be  no  deviation  at 
emergence,  but  at  the  other  incidence-point  Bl  the  ray  will  be  bent 
away  from  the  prism-edge.  And,  finally,  if  one  of  the  angles  of  the 
triangle  VB1B2  at  Bl  or  B2  is  obtuse,  for  example,  the  angle  at  B2, 
the  deviation  at  emergence  will,  it  is  true,  be  towards  the  prism-edge, 
but  this  will  not  be  so  great  as  the  previous  deviation  at  Bl  which  was 
away  from  the  edge ;  as  will  be  easily  seen  by  examining  a  diagram  for 
this  case.  So  that  in  every  case,  provided  n  >  I,  the  total  deviation 
will  be  away  from  the  prism-edge. 

If  n  <  i,  all  these  effects  are  reversed. 

ART.  22.     ANALYTICAL  INVESTIGATION  OF  THE  PATH  OF  A  RAY  REFRACTED 
THROUGH  A  PRISM  IN  A  PRINCIPAL  SECTION. 

70.  The  angles  of  incidence  and  refraction  at  the  first  and  second 
faces  of  the  prism,  denoted  by  alt  a(  and  a2,  a'2,  respectively,  are,  by 
definition  (§  14),  the  acute  angles  through  which  the  normal  to  the 
refracting  surface  at  the  incidence-point  has  to  be  turned  in  order  to 
bring  it  into  coincidence  with  the  incident  and  refracted  rays  at  each 
face  of  the  prism.  The  angle  of  deviation  or  the  total  deviation  of  a 
ray  refracted  through  a  prism,  denoted  by  the  symbol  €,  is  the  angle 
between  the  directions  of  the  emergent  and  incident  rays,  or  the  angle 
through  which  the  emergent  ray  must  be  turned  around  the  point  D 
(Fig.  32)  in  order  that  it  may  coincide  with  the  incident  ray  in  both 
position  and  direction.  Thus,  e  =  Z.JVG. 

Assuming  that  the  prism  is  surrounded  by  the  same  medium  on 
both  sides,  we  have  obviously  the  following  system  of  equations: 


n  =  n[/nlt     n.2  =  n^ 
sin  «   =  ft  -sin  a,     w-sin  «   =  sin 


(25) 


Combining  these  formulae,  we  obtain: 


sin  a2  =  sin  a^cos  /3  —  sin  /3-  i/n"  —  sin  c^;  (26) 

whence,  knowing  the  relative  index  of  refraction  (n  =  n'^n^  =  n[/n'2) 
and  the  refracting  angle  (/3)  of  the  prism,  we  can  compute  the  angle 
of  emergence  (a'2)  corresponding  to  any  given  value  of  the  angle  of 
incidence  (a^  at  the  first  face  of  the  prism. 

The  total  deviation  (e)  of  a  ray  refracted  through  a  prism  depends 


§  71.]  Refraction  Through  a  Prism  or  Prism-System.  81 

only  upon  the  values  of  the  magnitudes  alt  /3  and  n:  for  given  values 
of  these  magnitudes,  the  angle  e  will  be  uniquely  determined  by  for- 
mulae (25).  So  long  as  n  is  different  from  unity  and  /3  is  different  from 
zero,  the  value  of  e  cannot  be  zero.  On  the  other  hand,  to  each  value 
of  e  there  corresponds  always  two  values  of  the  angle  ax ;  for  a  second 
ray  incident  on  the  first  face  of  the  prism  at  an  angle  equal  to  the 
angle  of  emergence  of  the  first  ray  will  evidently  emerge  at  the  second 
face  at  an  angle  equal  to  the  angle  of  incidence  of  the  first  ray  at  the 
first  face;  and  hence  it  is  obvious  that  these  two  rays  will  undergo 
equal  deviations  in  traversing  the  prism.  For  example,  suppose  that 
the  values  of  the  angles  of  incidence  and  emergence  of  the  first  ray 
were  o^  =  0,  a2  =  6':  a  second  ray  incident  on  the  first  face  of  the 
prism  at  the  angle  aL  =  —  6'  will  emerge  at  the  second  face  at  an  angle 
a '2  =  —6,  and  both  of  these  rays  will  have  the  same  deviation,  viz., 
c  =  0  —  0'  —  |8. 

71.  Analytical  Investigation  of  the  Case  of  Minimum  Deviation. 
We  have  just  seen  that  there  is  always  a  pair  of  rays  for  which  the 
deviation  (e)  has  a  given  value.  One  pair  of  rays  for  which  the 
deviation  is  the  same  are  the  two  identical  rays  determined  by  the 
relation  : 

«!  =  0  =  —  a'2. 

In  this  case  the  course  of  the  ray  through  the  prism  is  symmetrical 
with  respect  to  the  two  faces  of  the  prism;  that  is,  the  ray  crosses 
at  right  angles  the  plane  which  bisects  the  dihedral  angle  0  between 
the  two  faces  of  the  prism. 

Inasmuch  as  the  deviation  (e)  is  a  symmetrical  function  of  o^  and 
—  a'21  it  must  be  either  a  maximum  or  a  minimum  when  the  ray  within 
the  prism  is  equally  inclined  to  both  faces  of  the  prism  (^  =  —  a'2). 
We  shall  show  that  as  a  matter  of  fact  the  deviation  in  this  case  is  a 
minimum. 

For  a  critical  value  of  the  angle  e,  we  must  have  de/d^  =  o.  Differ- 
entiating the  prism-formulae  above,  we  obtain: 

,  da,  da2  -   ,  da* 

cos  «!  =  n  -  cos  «L  — — ,     n  •  cos  a2  ~, —  =  cos  a2  -^ — 

da(       da~  de  da? 


da.^       da.^  dal 
These  latter  give: 

de  cos  al  -  cos  a2 

da*  cos  a(  -  cos  a, 


OF   TH€ 

UNIVERSITY 


82  Geometrical  Optics,  Chapter  IV.  [  §  71. 

and,  hence,  putting  dejd^  —  o,  we  have: 

cos  «!      cos  «2 
cos  a(      cos  a2 

Now  the  first  side  of  this  equation  is  a  function  of  o^  and  n,  whereas 
the  second  side  of  the  equation  is  the  same  function  of  a'2  and  n;  and 
therefore  we  must  have; 

«!  =  ±  «;. 

The  upper  sign  is  inadmissible  here,  as  the  value  ^  =  +  a'2  would 
make  the  refracting  angle  of  the  prism  equal  to  zero,  which  cannot  be 
in  the  case  of  a  prism.  Hence,  the  critical  value  of  the  angle  e  occurs 
when  we  have: 

«i  =  -  "2; 
and,  consequently,  also: 

«i  =  -  «2- 

As  we  saw  above,  this  was  the  value  of  al  when  the  ray  crossed  the 
prism  symmetrically. 

In  order  to  determine  whether  this  critical  value  of  c  is  a  maximum 
or  minimum,  we  shall  have  to  investigate  the  sign  of  the  second  deriva- 
tive of  €.  Differentiating  the  formula  above  for  the  first  derivative, 
we  obtain: 

2    /        2    '  d*e  >  /  /  •         da2        .  \ 

cos  «!  •  cos  «2  -r-j  =  cos  ai ' cos  a2 1  cos  ai ' sm  az  ~j \~  sm  ai ' cos  a2 ) 

cLo-i  \  da.^  / 

(/     .        /  da'2  ,      .          da(\ 

cos  «!  •  sm  a2  -j h  cos  a2  •  sm  a-^  -, —  J. 

Now  when  de/d^  =  o,  we  have: 

da2       dcti         cos  a2       da2  ,  , 

"3     =  "3     ==  >    "3     =i,    a\  =  —  a2     and     oji  =  —  (Xo: 

ac*!       aa       w-cos^     oaj 

moreover, 

<*[  =  ~  «2  =  18/2; 

hence,  substituting  these  values  in  the  above,  we  obtain  finally,  when 
de/dal  =  o: 

n-  cos  cvcos  a(  —2  =  (nz  —  i)  -sin  j8. 

CkCti 

Since  all  the  values  of  the  angles  c^  and  «J  are  comprised  between 
-f-  7T/2  and  —  7T/2,  and  since  the  angle  0  is  positive,  the  sign  of  d2e/dal 


§  72.]  Refraction  Through  a  Prism  or  Prism-System.  83 

depends  on  the  value  of  n.  If  n  >  I,  d2e/dal  will  be  positive,  and 
hence  for  ax  =  —  a'2  the  deviation  e  has  a  minimum  value.  But  if 
n  <  i,  d2e/dal  will  be  negative,  so  that  we  obtain  the  rather  unexpected 
result  that,  under  these  circumstances,  the  deviation  has  a  maximum 
value.  The  explanation  is  apparent;  for  if  we  recall  that  the  angle  e 
is  negative  when  n  <  I,  as  will  be  seen  by  an  inspection  of  Fig.  33, 
it  is  evident  that  a  maximum  value  of  e  in  this  case  corresponds  to  a 
minimum  absolute  value. 

If  the  critical  value  of  the  angle  e  is  denoted  by  the  symbol  €0,  we 
have,  therefore,  for  the  Position  of  Minimum  Deviation  the  following 
set  of  equations: 


(27) 


sin  - 

The  last  of  the  above  formulae  is  the  basis  of  the  FRAUNHOFER-method 
of  determining  the  refractive  index  n,  the  angles  /3  and  €0  being  capable 
of  easy  measurement. 

72.  Other  Special  Cases.  If  the  emergent  ray  is  normal  to  the 
second  face  of  the  prism,  we  must  put  az  =  a'2  =  o;  and,  thus,  for  the 
case  of  perpendicular  emergence  at  the  second  face,  we  have  ct(  =  /3, 
«!  =  |8  +  e,  so  that  we  obtain : 

sin  (0  +  e) 

sin/3       '     «2  =  "2  =  °- 

This  also  is  a  convenient  formula  for  the  experimental  determination 
of  the  value  of  the  refractive  index  n.  The  procedure  is  described  in 
treatises  on  physics. 

Case  of  a  Thin  Prism  (Prism  with  very  small  Refracting  Angle). 
If  the  refracting  angle  of  the  prism  is  so  small  that  we  may  put 
sin  0  =5=  /3,  cos  j3  =  i ,  the  deviation  e  will  also  be  a  small  angle  of  the 
same  order  of  magnitude.  In  this  case,  therefore,  since 

«;  =  <*!-  (]8  +  e), 
we  have: 

sin  a4  =  sin  e^  —  (0  +  e)  cos  a} . 


g4  Geometrical  Optics,  Chapter  IV.  [  §  72. 

Moreover,  since 

sin  «2  =  n-  sin  «2,     «2  =  a{  —  0, 
we  obtain 

sinaj  =  n  (sinaj  —  jS-cosaJ)  =  sin  a^  —  n-ft-cosai. 

Therefore,  equating  these  two  values  of  sin  a'2,  we  obtain  in  the  case 
of  finite  value  of  the  incidence-angle  aL: 


/     cos  a[         \ 
=  01  n-     --  i     . 
\    cos «!         / 


In  case  the  angle  of  incidence  ^  is  also  a  very  small  angle,  we  obtain 
the  following  approximate  formula  for  the  deviation: 

e  =  (n  -  i)j8.  (28) 

In  these  formulae  the  angles  are  all  measured  in  radians.  According 
to  (28),  for  small  values  of  both  /3  and  alt  the  deviation  e  is  propor- 
tional to  the  refracting  angle  /?,  and  is  independent  of  the  incidence- 
angle  aL. 

The  Case  of  Total  Reflexion  at  the  Second  Face  of  the  Prism.  If 
the  angle  of  emergence  at  the  second  face  of  the  prism  is  a  right  angle, 
that  is,  if  a'?,  =  —  90°,  the  emergent  ray  B2Q'2  will  proceed  along  (or 
"graze")  the  second  face  n2.  When  this  occurs,  we  have  a2  =  —  A, 
where  the  symbol  A,  defined  by  the  formula: 

n,       i 

sin  A  —  —  =  -  ,     (n  >  i), 
nl      n 

denotes  the  magnitude  of  the  so-called  "critical  angle"  (§  27)  for  the 
two  media  whose  relative  index  of  refraction  =  n.  If  the  absolute 
value  of  the  angle  a2  is  greater  than  this  critical  angle  A,  the  ray  will 
be  totally  reflected  at  the  second  face  of  the  prism,  and  there  will  be 
no  emergent  ray.  We  proceed  to  discuss  this  case  in  some  detail. 
For  a  prism  of  given  refracting  angle  (/3),  there  is  a  certain  limiting 
value  i  of  the  angle  of  incidence  ax  at  the  first  face  of  the  prism  for 
which  we  shall  have  a2  =  -  A  and  a'z  =  —  90°;  so  that  a  ray  which 
is  incident  on  the  first  face  of  the  prism  at  an  angle  less  than  this  limiting 
angle  t  will  not  pass  through  the  prism,  but  will  be  totally  reflected  at  the 
second  face.  Putting  av  =  t,  a2  =  A  in  formulae  (25),  we  obtain  the 
formula : 

sin  i  =  n-  sin  (/3  —  A) ;  f  (29) 

whereby  the  limiting  angle  of  incidence  (t)  for  a  given  prism  can  be 


72.] 


Refraction  Through  a  Prism  or  Prism-System. 


85 


computed.     Examining  this  formula,  we  derive  the  following  con- 
clusions : 

(1)  If    /3  >  2 A,  then    (since   sin  A  =  i/ri)    sin  t   is    greater  than 
unity;    which   means   that   for   such   a   prism   there  is  no  limiting 
angle  i.     Accordingly,  if  the  refracting  angle  of  the  prism  is  more  than 
twice  as  great  as  the  "critical  angle"  (A),  it  will  be  impossible  for  any 
ray  whatever  to  be  transmitted  through  the  prism.     For  instance,  for  a 
crown-glass  prism  in  air,  the  angle  A  =  40°  50',  and  hence  a  prism  of 
this  material  with  a  refracting  angle  greater  than  81°  40'  will  not  per- 
mit any  ray  to  emerge  at  its  second  face. 

(2)  If  |8  =  2 Ay  we  obtain,  by  formula  (29),  t  =  90°.     In  this  case 
the  limiting  ray  IlBl  (Fig.  36)  will  "graze"  the  first  face  of  the  prism. 


FIG.  36. 

REFRACTING  ANGLE  OF  PRISM  EQUAL  TO 
TWICE  THE  CRITICAL  ANGLE.  The  only  ray 
that  can  pass  through  the  prism  is  I\B\B&.z. 


FIG.  37. 

REFRACTION  OF  A  RAY  THROUGH  A  PRISM. 
limiting   Ray   in    the   case   when    2A  >  ft  >  A 


This  is  the  only  ray  that  can  pass  through  the  prism,  and  it  will  emerge 
at  B2  and  proceed  along  the  second  face  of  the  prism  in  the  direction 
.#2/z2.  Since  here  we  have  ^  =  i  =  90°  =  —  a'2,  evidently  this  is  also 
the  ray  of  minimum  deviation  (§71)  for  this  prism  (€0  =  /3). 

(3)  If  the  refracting  angle  0  is  greater  than  A,  but  less  than   2 A 
(that  is,  2 A  >  0  >  A),  the  value  of  t,  as  determined  by  (29),  will  be 
comprised  between  90°  and  o°.     Hence,  for  a  prism  with  a  refracting 
angle  such  as  this,  the  limiting  ray  IlBl  will  have  a  direction  between 
the  directions  nlBl  and  N^;  that  is,  the  Z.VB1I1  (Fig.  37)  will  be 
an  obtuse  angle. 

(4)  If  )8  =  Aj  we  find  t  =  o;  so  that  for  such  a  prism  the  limiting 
incident  ray  I1Bl  will  be  in  the  direction  of  the  normal  at  Bl  (Fig.  38). 


86 


Geometrical  Optics,  Chapter  IV. 


[§72. 


In  this  case,  therefore,    Z  VB^  =  90°.     The  ray  which  "grazes"  the 
first  face'of  this  prism  will  meet  the  second  face  normally. 

(5)  Finally,  if  /3  <  A,  formula  (29)  gives  in  this  case  a  negative  value 
of  the  angle  i;  and,  hence,  for  a  prism  with  such  a  refracting  angle  the 


FIG.  38. 

REFRACTION  OF  A  RAY  THROUGH  A 
PRISM.  Case  when  ft  =  A  (n  >  l).  Limit- 
ing incident  ray  meets  first  face  of  prism 
normally. 


FIG.  39. 

PATH  OF  LIMITING  RAY  FOR  PRISM 
OF  REFRACTING  ANGLE  ft  <  A  (n  >  1). 


limiting  incident  ray  will  lie  on  the  same  side  of  the  normal  at  Bl  as 
the  vertex  V  of  the  prism,  so  that  now    Z  VB1I1  is  an  acute  angle 

(Fig.  39). 

In  all  these  cases  incident  rays  which  meet  the  first  face  V^  of 
the  prism  at  the  point  Blt  and  which  are  comprised  within  the  angle 
/!#!/*!  will  be  transmitted  through  the  prism;  whereas  all  rays  inci- 
dent at  Bl  and  lying  within  the  angle  VB1I1  will  be  totally  reflected 
at  the  second  face  of  the  prism. 

If  ft  =  o°,  the  prism  is  a  Slab  with  Parallel  Faces,  and  then  we  have 
t  =  —  90°  and  Z  VB1I1  =  o°.  All  incident  rays  will  be  transmitted 
through  such  a  slab. 

In  KOHLRAUSCH'S  method  of  measuring  the  relative  index  of  re- 
fraction (n),  the  prism  is  adjusted  so  that  the  incident  ray  "grazes" 
the  first  face  of  the  prism;  in  which  case  the  value  of  n  will  be  given 
by  the  formula: 

/-= —          cos  /3  —  sin  ai 
V n*  -  I  =  —-2 -.     (a,  =  00°). 


The  Total-Reflexion  Principle  is  made  use  of  also  in  the  so-called 
Total  Refractometers  of  ABBE  and  PULFRICH  for  the  determination  of 
the  refractive  index. 


73.] 


Refraction  Through  a  Prism  or  Prism-System. 


87 


73.  For  convenience  of  reference  the  following  collection  of  formulae 
for  calculating  the  path  of  a  ray  refracted  through  a  prism  in  a  prin- 
cipal section  are  placed  here. 

PRISM  FORMULAE. 


sin  «!  =  cos  j8  •  sin  a'2  +  sin  /3     nz  —  sin2  a'2  ; 
sin  «2  =  cos  j3  •  sin  o^  —  sin  /3  1/w2  —  sin2  «L  ; 


Minimum  Deviation: 


Grazing  Incidence: 


,  ft 

:1  =  w-sm-;      < 


ai  —  9°°»     sin  «2  =  cos  j8  ~  sin  £  1/w2  —  i  ; 
a(  =  A,     a2  =  A  —  0,     e  =  90°  —  a'2  —  |8. 
Grazing  Emergence: 

a'2=  —  90°,     sin  «!  =  sin  |8  Vn2  —  i  —  cos  0; 
a[=p-A,     a2=-A,     e  =  90°  +  «i  —  /3. 
Normal  Incidence: 

at  =  o,     sin  «2  =  ~~  w  '  sin  0  ; 
ai  =  o,     a2  =  -  j8,     e  =  0  -  c4 
Normal  Emergence: 

a'2  =  o,     sin  ojj  =  w  •  sin  |S  ; 


=  o, 


e  =  «   — 


The  subjoined  table  gives  the  results  of  the  calculations  of  the 
values  of  these  angles  for  a  prism  of  flint  glass,  designated  in  the  glass 
catalogue  of  SCHOTT  u.  Gen.,  Jena,  as  0.103,  the  refractive  index  of 
which  for  rays  of  light  corresponding  to  the  FRAUNHOFER  ZMine  is 
n  =  1.620  2.  The  refracting  angle  of  the  prism  is  taken  as  30°. 
The  value  of  the  critical  angle  A  (=  sin"1  ifn)  is  38°  6'  45". 


First  Face. 

Second  Face. 

Deviation. 

e 

Angle  of 
Incidence. 
<M 

Angle  of 
Refraction. 
ai' 

Angle  of 
Incidence. 

«2 

Angle  of 
Refraction. 
02' 

90°    o'    o" 
54      6  20 
24   47  34 

000 

—13    13     i 

38°  6'  45" 
30   o     o 
15    o     o 

000 

-8   6  45 

8°  6'  45" 

000 

—15    o     o 
—  30   o     o 
-38   6  45 

13°  13'    l" 

000 

—24   47  34 
—54     6  20 
—  90     o     o 

46°  46'  59/x 
24     6  20 
19   35     8 
24      6  20 
46   46  59 

88 


Geometrical  Optics,  Chapter  IV. 


[§74. 


ART.  23.  PATH  OF  A  RAY  REFRACTED  ACROSS  A  SLAB  WITH  PARALLEL  FACES. 

74.  If  the  two  plane  refracting  surfaces  of  the  prism  are  parallel 
(0  =  0),  we  no  longer  call  it  a  prism,  but  a  slab  or  plate  with  parallel 
faces.  The  path  of  a  ray  refracted  across  such  a  slab  may  evidently 
be  constructed  as  follows: 

Around  the  incidence-point  B\  (Fig.  40),  where  the  incident  ray 
meets  the  first  surface  of  the  slab,  describe  in  the  plane  of  incidence 


/<* 

FIG.  40. 
CONSTRUCTION  OF  PATH  OF  RAY  REFRACTED  ACROSS  A  SLAB  WITH  PLANE  PARALLEL  FACES. 

three  concentric  circles  of  radii  r,  n^/n'^  n'2r/n'lt  where  the  radius  r 
has  any  arbitrary  length,  and  where  nlt  n{  and  n'2  denote  the  absolute 
indices  of  refraction  of  the  first,  second  and  third  medium,  respectively. 
Let  G  designate  the  point  where  the  incident  ray  L^B^  meets  the  circle 
of  radius  n^r/n'^  and  draw  GE  perpendicular  to  the  first  face  of  the 
slab  at  the  point  £,  and  let  this  perpendicular,  produced  if  necessary, 
meet  the  circle  of  radius  r  in  the  point  H\  then  the  straight  line  HBl 
will  evidently  give  the  direction  of  the  ray  after  refraction  at  the  first 
face.  Moreover,  if  J  designates  the  point  where  the  straight  line  GE 
meets  the  circle  of  radius  n2rjn{,  and  if  B2  designates  the  incidence- 
point  of  the  ray  at  the  second  face  of  the  slab,  the  straight  line  B2Q'2 
drawn  parallel  to  JBl  will  give  the  direction  and  path  of  the  emergent 
ray. 

In  the  special  case  when  the  last  medium  is  identical  with  the  first, 


§  75.]  Refraction  Through  a  Prism  or  Prism-System.  89 

so  that  we  have  n'2  =  n^  two  of  the  circles  used  in  the  above  con- 
struction will  coincide,  and  accordingly  the  points  designated  by  G 
and  /  will  be  coincident.  And  hence  in  this  case  the  emergent  ray 
will  be  parallel  to  the  incident  ray.  The  perpendicular  distance  be- 
tween the  parallel  paths  of  the  incident  and  emergent  rays  is  equal  to 

sin  (a  —  a') 

— "i d, 

cos  a' 

where  d  denotes  the  thickness  of  the  slab  and  a,  a'  denote  the  angles 
of  incidence  and  refraction  at  the  first  face. 

75.  We  may  also  investigate  here  the  path  of  a  ray  which  is  refracted 
in  succession  at  a  series  of  parallel  refracting  plane  surfaces  ^  n2,  //3,  etc. 
If  «j,  n(j  #2,  etc.,  denote  the  absolute  indices  of  refraction  of  the  media 
traversed  in  succession  by  the  ray,  and  if  alt  a[ ;  a2,  a'2 ;  etc.,  denote  the 
angles  of  incidence  and  refraction  at  the  series  of  parallel  refracting 
planes,  then,  supposing  that  we  have,  say,  m  such  planes,  we  shall 
have  the  following  set  of  equations: 

Wj-sin  <*!  =  wj-sin  aj, 
n[  •  sin  a2  =  n'2-  sin  az, 


nm_vsmam  =  nm-smam. 

Multiplying  together  the  expressions  on  each  side  of  these  equations, 
and  remarking  that,  since  the  refracting  planes  are  parallel,  we  must 
have: 


we  obtain  immediately: 

Wj-sin  «!  =  n'm-sin  am. 

Thus,  without  knowledge  of  either  the  number  or  the  nature  of  the 
intervening  media,  this  formula  enables  us  to  find  the  direction  am  of 
the  ray  which  emerges  into  the  last  medium.  The  effect  of  the  inter- 
vening media  between  the  first  and  the  last  is  merely  to  produce  a 
parallel  displacement  of  the  emergent  ray;  otherwise,  everything  is 
the  same  as  if  the  ray  had  been  refracted  from  the  first  to  the  last 
medium  across  a  single  refracting  plane.  If  the  last  medium  is  iden- 
tical with  the  first,  so  that  we  have  n'm  =  n^  then  a^  =  «i,  and  the 
direction  of  the  emergent  ray  will  be  parallel  to  that  of  the  incident 
ray,  as  we  saw  above  in  the  case  of  a  slab  surrounded  by  the  same 


90 


Geometrical  Optics,  Chapter  IV. 


[§76. 


medium  on  both  sides.  For  example,  if  we  interpose  in  front  of  the 
object-glass  of  a  telescope  pointed  towards  a  star  a  plate  of  glass  with 
plane  parallel  sides,  the  image  of  the  star  will  not  be  deviated  thereby. 
This  fact  is  employed  in  a  simple  method  of  testing  with  a  high  degree 
of  precision  whether  or  not  two  faces  of  a  plate  of  glass  are  accurately 
parallel. 

ART.  24.     REFRACTION,  THROUGH  A  PRISM,  OF  AN  INFINITELY  NARROW 

HOMO  CENTRIC  BUNDLE  OF  INCIDENT  RAYS,  WHOSE  CHIEF  RAY 

LIES  IN  A  PRINCIPAL  SECTION  OF  THE  PRISM. 

76.  If  an  infinitely  narrow  homocentric  bundle  of  incident  rays 
is  refracted  through  a  prism,  and  if  the  chief  ray  lies  in  the  plane  of 
a  principal  section  of  the  prism,  the  meridian  sections  of  the  incident 
and  refracted  bundles  of  rays  will  lie  in  the  principal  section  which 
contains  the  chief  rays  of  the  bundles,  and  which  is  the  plane  of  inci- 
dence of  the  chief  incident  ray  uv ;  whereas  the  planes  of  the  sagittal 
sections  of  the  bundles  of  incident  and  refracted  rays  will  intersect  in 
straight  lines  parallel  to  the  edge  of  the  prism. 

We  give,  first,  the  construction  of  the  I.  and  II.  Image-Points  corre- 


FIG.  41. 

REFRACTION,  THROUGH  A  PRISM,  OF  AN  INFINITELY  NARROW  BUNDLE  OF  RAYS  WHOSE  CHIEF 
RAY_LIES  IN  A  PRINCIPAL  SECTION  OF  THE  PRISM.  Construction  of  the  I.  and  II.  Image-Points 
St',  St'  on  the  chief  emergent  ray  us'  corresponding  to  Object-Point  Si  on  chief  incident  ray  MI. 

sponding  to  a  homocentric  object-point.    Let  Sl  (Fig.  41)  be  the  radiant 
point  or  homocentric  object-point  of  the  bundle  of  incident  rays,  the 


§  76.]  Refraction  Through  a  Prism  or  Prism-System.  91 

chief  ray  of  which  (u^  is  incident  on  the  first  face  of  the  prism  at  the 
point  B^  The  path  of  this  ray  within  the  prism  and  after  emergence 
from  it  is  constructed  by  REUSCH'S  Construction  (§  66).  We  shall 
assume  that  the  medium  of  the  emergent  rays  is  identical  with  that  of 
the  incident  rays,  so  that  n  =  n{  jn^  =  n(  /n'2.  In  the  figure VG  =  VZ{  jn. 
The  straight  line  VG  is  drawn  through  V  parallel  to  the  chief  inci- 
dent ray  u±\  the  straight  line  GZ[  is  drawn  through  G  normal  to  the 
first  face  ^  of  the  prism.  The  path  within  the  prism  of  the  chief 
ray  u(  of  the  astigmatic  bundle  of  rays  refracted  at  the  first  face  will 
be  along  the  straight  line  B1B2  drawn  from  Bl  parallel  to  the  straight 
line  VZ[.  And  if  Z[J  is  normal  to  the  second  face  of  the  prism,  the 
chief  ray  u'2  of  the  astigmatic  bundle  of  emergent  rays  will  be  along 
the  straight  line  B2u'2  parallel  to  VJ. 

On  the  ray  VG  which  is  incident  on  the  first  face  of  the  prism  at 
the  point  V  where  the  principal  section  intersects  the  edge  of  the 
prism  and  which,  by  construction,  is  parallel  to  the  incident  ray  #t 
there  is  a  point  Zx  to  which  the  point  Z{  on  the  ray  VZ{  resulting 
from  the  refraction  of  the  ray  VG  at  the  first  face  of  the  prism,  cor- 
responds as  I.  Image-Point.  This  point  Zl  may  be  constructed  accord- 
ing to  the  second  of  the  two  constructions  given  in  §  64,  as  follows: 
Through  Z(  draw  a  straight  line  perpendicular  to  FZj,  and  let  Ult  U2 
designate  the  positions  of  the  two  points  where  this  straight  line  meets 
the  prism-faces  j^,  ju2,  respectively.  Let  X  designate  the  point  of 
intersection  of  the  normal  at  V  to  the  first  face  of  the  prism  with  the 
straight  line  Ufi.  The  point  ZL  is  the  foot  of  the  perpendicular  let 
fall  from  X  on  the  incident  ray  VG.  In  the  same  way  the  I.  Image- 
Point  Zj  corresponding  to  the  point  Z(  on  the  ray  VZ(  incident  on 
the  second  face  of  the  prism  at  V  will  be  found  to  lie  on  the  emergent 
ray  VJ  at  a  point  which  is  determined  by  drawing  UJ  to  meet  at 
F  the  normal  at  V  to  the  second  face  of  the  prism,  and  dropping 
from  F  a  perpendicular  on  VJ,  the  foot  of  which  will  be  the  required 
point  Z'2. 

Hence,  according  to  the  relations  of  affinity  which  were  shown  in 
§  63  to  exist  between  the  object-points  and  the  image-points  in  the 
case  of  the  refraction  of  parallel  rays  at  a  plane  surface,  the  I.  Image- 
Point  S(  corresponding  to  the  homocentric  object-point  Sl  on  the  chief 
ray  u^  of  the  bundle  of  incident  rays  will  be  the  point  of  intersection 
of  the  straight  line  drawn  through  5X  parallel  to  Z^,  with  the  chief 
ray  u(  of  the  astigmatic  bundle  of  rays  refracted  at  the  first  face  of 
the  prism.  This  point  S{  is  the  vertex  of  the  pencil  of  meridian  rays 
after  refraction  at  the  first  face  of  the  prism.  Considered  with  re- 


92  Geometrical  Optics,  Chapter  IV.  [  §  77. 

spect  to  the  refraction  at  the  second  face  of  the  prism,  this  point  is 
the  vertex  of  the  pencil  of  meridian  rays  which  are  incident  on  this 
face;  so  that  it  might  also  be  designated  as  the  point  S2.  From  S[ 
draw  a  straight  line  parallel  to  Z[  Z2  meeting  the  emergent  chief  ray 
«2  in  the  point  S2,  which  is  accordingly  the  vertex  of  the  pencil  of  emer- 
gent meridian  rays  of  the  bundle,  and  which  is  therefore  the  I.  Image- 
Point  on  the  emergent  chief  ray  u'2  corresponding  to  the  object-point 
5t  on  the  chief  incident  ray  ur 

Again,  the  normal  to  the  first  face  of  the  prism  drawn  through  the 
object-point  Sl  on  the  chief  incident  ray  u^  will  meet  the  chief  ray 
u(  of  the  astigmatic  bundle  of  rays  refracted  at  this  face  in  the  II. 
Image-Point  S{ ;  which  is  the  vertex  of  the  pencil  of  sagittal  rays  after 
refraction  at  the  first  face  of  the  prism.  This  point  may  also  be  desig- 
nated as  the  point  S2  by  regarding  it  as  the  vertex  of  the  pencil  of 
sagittal  rays  which  are  incident  on  the  second  face  of  the  prism.  And, 
finally,  if  through  S(  we  draw  a  normal  to  the  second  face  of  the  prism, 
this  normal  will  meet  the  chief  emergent  ray  u'2  in  the  point  5^  which 
is  the  II.  Image-Point  on  the  chief  emergent  ray  u'2  corresponding  to 
the  object-point  5,  on  the  chief  incident  ray  ulf 

Applying  here  the  results  of  §  63,  we  can  say: 

Corresponding  to  a  range  of  homocentric  object-points  Plt  Qly  Rly 
on  an  incident  chief  ray  ult  which  is  refracted  through  a  prism  in  a  prin- 
cipal section,  we  have  a  similar  range  of  I.  Image- Points  P2J  Q2,  R2, 
and  a  similar  range  of  II.  Image-Points  P2,  ~Q2,  R2,  - . .  both  lying  on  the 
emergent  chief  ray  u2. 

77.  Formulas  for  Calculation  of  the  Positions  on  the  Chief  Emer- 
gent Ray  of  the  I.  and  II.  Image-Points.  Let  a,,  a(  and  a,,  a'2  denote 
the  angles  of  incidence  and  refraction  of  the  chief  ray  of  the  bundle 
at  the  first  and  second  faces  of  the  prism,  respectively;  so  that  if  nlf 
n[  and  n'2  denote  the  absolute  indices  of  refraction  of  the  three  media 
traversed  by  the  ray  in  succession,  we  shall  have: 

f    •      t  .  /  t         , 

nl •  sin  <*!  =  n± •  sm  alt     n2- sm  <x2  =  n^-  sin  a2. 

Referring  to  the  figure  (Fig.  41),  let  us  employ  the  following  symbols: 
B&  =  slt     BpS;  =  s[,     B&  =  s[,     B2S2  =  s'2,     B2S2  =  s'2; 

then,  by  formulae  (19)  and  (21),  we  have: 
For  the  Meridian  Rays: 

n[-cosza'  w'-cos*a' 


§  78.]  Refraction  Through  a  Prism  or  Prism-System.  93 

and  for  the  Sagittal  Rays: 


If  we  put  B^z  =  6L,  then: 

B.2Sl  =  B2Bl  +  BlSl  =  sl  —  dlt 
B2S[  =  BZB1  +  B&  =  s[  -5i; 

and  substituting  these  values  in  the  equations  above,  and  eliminating 
s[  and  ~s[j  we  obtain  finally  the  following  formulae  for  determining  the 
positions  of  the  I.  and  II.  Image-Points  on  the  chief  emergent  ray: 
Meridian  Rays: 

,       n'z  cos2 


/    \ 

—  i--i  (30) 

COS     (X2       COS 


Sagittal  Rays: 


Thus,  knowing  the  position  of  the  object-point  5X  on  the  chief  inci- 
dent ray  wx  of  the  homocentric  bundle  of  incident  rays,  and  knowing 
also  the  optical  and  geometrical  constants  of  the  prism,  we  can  cal- 
culate by  means  of  formulae  (30)  and  (31)  the  positions  of  the  I.  and 
II.  Image-Points  S2  and  32  on  the  chief  ray  u2  of  the  astigmatic  bundle 
of  emergent  rays. 

78.  Convergence-Ratios  of  the  Meridian  and  Sagittal  Rays.  Let 
Gl  and  Jl  (not  shown  in  the  figure)  designate  the  positions  of  two 
points  on  the  first  face  of  the  prism  infinitely  near  to  the  incidence- 
point  B^:  the  point  G}  lying  in  the  plane  of  the  meridian  section  and 
the  point  Jl  in  the  plane  of  the  sagittal  section  of  the  infinitely  narrow 
homocentric  bundle  of  incident  rays  which  emanate  from  the  object- 
point  Sp  The  straight  lines  S^G^  SlJl  will  be  the  paths  of  a  second- 
ary meridian  ray  and  a  secondary  sagittal  ray  of  the  bundle  of  inci- 
dent rays.  Also,  let  G2,  J2  designate  the  points  where  these  two  rays, 
after  refraction  at  the  first  face  of  the  prism,  meet  the  second  face. 
If  the  angles  which  these  secondary  incident  rays  make  with  the  chief 
incident  ray  u^  and  the  angles  which  the  corresponding  secondary 
emergent  rays  make  with  the  chief  emergent  ray  u'2  are  denoted  as 
follows  : 

Z  B^fa  =  dal9     Z  B.S.J,  =  rf\lf     Z  B2S2G2  =  da'2,     Z  B2S2J2  =  d\'21 

the  ratios: 

7-^*2       -^  _  <^2 
ZM  ~  da,  '        M  ~  dUS 


94  Geometrical  Optics,  Chapter  IV.  [  §  80. 

are  called  the  Convergence-  Ratios  of  the  Meridian  and  Sagittal  Rays, 
respectively.     Applying  formulae  (20)  and  (22)  to  the  refractions  at 
the  two  faces  of  the  prism,  we  obtain  immediately: 
Meridian  Rays: 

da'2      #!  cos  o^  •  cos  a2  . 
u  ~  dct^  ~  n'2  cos  a(  •  cos  a'2  ' 
Sagittal  Rays: 


79.  If  the  prism  is  surrounded  by  the  same  medium  on  both  sides, 
so  that  n'2  =  wlf  the  formulae  above  (30),  (31),  (32)  and  (33)  may  be 
simplified  by  putting  n  =  n'^n^  =  n[/n'2.     In  this  case  the  convergence- 
ratio  of  the  sagittal  rays  will  be  equal  to  unity  for  all  directions  of 
the  chief  incident  ray.     Moreover,  if  when  n'2  =  n^  the  chief  incident 
ray  has  the  direction  of  the  ray  of  minimum  deviation,  so  that  (§71) 
cos  «!  •  cos  ct2  =  cos  a(  •  cos  a'2,  the  convergence-ratio  of  the  meridian 
rays  will  likewise  be  equal  to  unity;  that  is,  ZM)0  =  I. 

In  general,  therefore,  the  image  of  a  luminous  point  as  seen  through 
a  prism,  viewed  either  by  the  naked  eye  or  through  a  telescope,  will  not 
be  a  point.  Depending  on  how  the  eye  or  telescope  is  focussed,  the  im- 
age of  a  point-source  of  light  will  appear  through  the  prism  as  a  small 
straight  line  parallel  to  the  prism-edge  (I.  Image-Line),  or  a  disc  of 
light,  or,  finally,  a  small  straight  line  lying  in  the  plane  of  the  prin- 
cipal section  of  the  prism  (II.  Image-Line).  See  description  of  L. 
BURMESTER'S  Experiment,  §  85.  In  a  prism-spectroscope  the  source  of 
light  is  usually  a  narrow  illuminated  slit  with  its  length  parallel  to 
the  prism-edge.  If,  as  is  usually  done  in  this  case,  we  focus  the  tele- 
scope on  the  II.  Image-Line,  the  slit-image,  except  near  its  ends,  will 
be  clear  and  distinct,  so  that  here  we  encounter  practically  no  serious 
disadvantage  on  account  of  astigmatism  (§86). 

80.  The  Astigmatic  Difference.     If  the  bundle  of  incident  rays  is 
itself  astigmatic,  instead  of  a  homocentric  object-point,  we  shall  have 
a  I.    Object-Point  Sl  and  a  II.  Object-Point  S^  and  the  astigmatic 
difference  (see  §  61)  of  the  bundle  of  incident  rays  will  be: 


where  sl  =  B^^  =  B^;  and  the  astigmatic  difference  of  the  cor- 
responding bundle  of  emergent  rays  will  be  : 

S2S2  =  s2  -  s'2. 


§  81.]  Refraction  Through  a  Prism  or  Prism-System.  95 

Accordingly,  from  formulae  (30)  and  (31),  we  obtain  the  following 
general  formula  for  the  astigmatic  difference  of  the  emergent  rays: 


2        '  2 

COS    «T  •  COS 

cos2  a, -cos2 


«i          n*          n'      /cos2  a'2        \ 

•^1 s^  —  — f  Oi  I  o —   I  l« 

«2  Wl  Wi      *  \COS     «2  / 


In  case  the  bundle  of  incident  rays  is  homocentric  (sl  —  sl  =  o),  the 
astigmatic  difference  of  the  bundle  of  emergent  rays  is  given  by  the 
following  formula: 

n^/Wcycos2^         \          n'2  (cos2  a'2         \ 

S2S2   =    —  I     -  z~         ~2  -   —    I    IS,    —   —  I   --  2  -      —    I   JOV  (34) 

#!  \cos  cvcos    a2        )  l      HI  \cos  a2         )  l 


According  to  this  formula,  therefore,  the  magnitude  of  the  astigmatic 
difference  of  the  bundle  of  emergent  rays  depends  not  only  on  the 
direction  of  the  chief  ray  of  the  homocentric  bundle  of  incident  rays, 
but  also  on  the  length  of  the  ray-path  B1B2  =  dl  within  the  prism. 
Moreover,  for  a  given  incident  chief  ray  ult  the  astigmatic  difference 
S2S2  depends  on  the  position  on  u^  of  the  homocentric  object-point  5t; 
for  it  increases  in  proportion  as  sl  increases;  that  is,  for  a  given  prism 
and  a  given  incident  chief  ray  ult  the  astigmatic  difference  of  the  bun- 
dle of  emergent  rays  is  proportional  to  the  distance  of  the  homocentric 
object-point  5X  from  the  incidence-point  Br 

81.  Magnitude  of  the  Astigmatic  Difference  in  Certain  Special 
Cases. 

(i)  If  the  object-point  Sl  on  the  chief  incident  ray  uv  coincides  with 
the  incidence-point  B^  at  the  first  face  of  the  prism,  that  is,  if  sl  =  o, 
the  astigmatic  difference  of  the  bundle  of  emergent  rays  will  have  the 
value  given  by  the  following  expression  : 


where  B2,  B2  designate  the  I.  and  II.  Image-Points,  respectively,  cor- 
responding to  a  homocentric  object-point  B1  coinciding  with  the  inci- 
dence-point of  the  chief  ray  u±  at  the  first  face  of  the  prism.  Evi- 
dently, if  the  incidence-point  Bl  is  regarded  as  an  object-point  on  «lf 
the  I.  and  II.  Image-Points  B[,  B[  on  the  chief  ray  u(  of  the  bundle 
of  rays  refracted  at  the  first  face  of  the  prism  will  coincide  with  each 
other  at  the  point  Br  If,  therefore,  through  Bv  we  draw  two  straight 
lines,  one  parallel  to  Z(Z'2  (see  Fig.  42)  and  one  perpendicular  to  the 
second  face  of  the  prism,  these  two  straight  lines  will  determine  by 
their  intersections  with  the  chief  emergent  ray  u2  the  I.  Image-Point 


96  Geometrical  Optics,  Chapter  IV.  [  §  81. 

B'2  and  the  II.  Image-Point  B'z,  respectively,  corresponding  to   the 
homocentric  object-point  Bl  on  ur 

(2)  If  the  object-point  is  a  point  Zl  lying  on  an  incident  chief  ray 
zl  which  meets  the  refracting  edge  of  the  prism,  so  that  the  ray  goes 
through  the  point  V  in  the  principal  section  of  the  prism,  in  this 
limiting  case  the  ray-length  within  the  prism  is  vanishingly  small,  and, 
hence,  putting  5X  =  o  in  formula  (34),  we  find: 


a'2        \ 

~~    I    I 

a2         ) 


•vzv 

(3)  The  condition  that  the  astigmatic  difference  shall  be  indepen- 
dent of  the  distance  of  the  homocentric  object-point  Sl  from  the  point 
B^  where  the  chief  incident  ray  meets  the  first  face  of  the  prism,  that 
is,  the  condition  that  the  astigmatic  difference  shall  be  independent 
of  the  magnitude  slt  is  evidently: 

cos  a(  •  cos  «2  =  cos  «r  cos  a2; 

which,  in  the  general  case,  leads  to  an  equation  of  the  eighth  degree  for 
calculating  the  value  of  the  angle  of  incidence  ^  in  order  to  ascertain 
what  must  be  the  direction  of  the  chief  incident  ray.  In  the  special 
case,  however,  when  the  prism  is  surrounded  on  both  sides  by  the 
same  medium  (n'2  =  n^),  the  equation  above  will  be  recognized  as 
the  condition  that  the  ray  shall  traverse  the  prism  with  minimum 
deviation  (§71).  Accordingly,  in  case  the  chief  ray  u0i  l  of  the  homo- 
centric  bundle  of  incident  rays  has  the  direction  of  the  ray  of  mini- 
mum deviation,  we  have  here  the  following  special  formulas: 
Minimum  Deviation  (n  =  n( \n±  —  n{  fn'2) : 


In  this  special  case,'  since  2a2  +  /3  =  o,  the  formula  for  the  magnitude 
of  the  astigmatic  difference  in  the  case  of  minimum  deviation  may  be 
written  also  in  the  following  form: 


n 


which  shows  clearly  that  the  magnitude  of  the  astigmatic  difference, 
which  in  every  case  depends  on  the  length  of  the  ray-path  within  the 
prism,  is  in  the  special  case  of  minimum  deviation  of  the  chief  ray 
directly  proportional  to  this  magnitude  5^  The  nearer  to  the  edge 


§  82.]  Refraction  Through  a  Prism  or  Prism-System.  97 

of  the  prism  the  chief  ray  u'0,  i  is,  the  smaller  will  be  the  astigmatic 
difference;  and  in  the  limiting  case  when  the  chief  incident  ray  co- 
incides with  the  ray  z0f  i,  which  is  the  ray  of  this  system  of  parallel 
incident  rays  that  meets  the  refracting  edge  of  the  prism,  the  astig- 
matic difference  vanishes  altogether,  so  that  the  I.  and  II.  Image- 
Points  Zj>2,  Zi|2  corresponding  to  any  object-point  Z0fl  on  the  chief 
incident  ray  z0t  i  are  coincident,  as  is  also  evident  from  the  formula 
given  under  (2)  above. 

These  results  may  be  stated  as  follows: 

To  every  actual  object-point  lying  on  an  incident  chief  ray,  which 
traverses  a  prism  in  its  principal  section  in  a  direction  such  that 

cos  otj-cos  a(  =  cos  «2-cos  a'2, 

there  corresponds  an  astigmatic  difference  of  the  astigmatic  bundle  of 
emergent  rays  which  is  independent  of  the  position  of  the  object-point 
on  the  incident  chief  ray,  and  which  is  proportional  to  the  distance  of 
the  chief  ray  from  the  refracting  edge  of  the  prism.  In  particular,  on 
the  incident  chief  ray  which  meets  the  refracting  edge  of  the  prism,  to 
every  object-point  there  corresponds  on  the  emergent  ray  a  homocentric 
image-point.  (This  latter  statement,  however,  has  merely  geometric 
significance,  since  in  this  limiting  case  the  length  of  the  ray-path 
within  the  prism  vanishes.)1 

(4)  The  special  case  when  the  astigmatic  difference  =  o  will  be 
considered  at  length  in  the  following  article  (Art.  25). 

ART.  25.     HOMOCENTRIC  REFRACTION,  THROUGH  A  PRISM,  OF  NARROW, 

HOMOCENTRIC  BUNDLE  OF  INCIDENT  RAYS,  WITH  ITS  CHIEF  RAY 

LYING  IN  A  PRINCIPAL  SECTION  OF  THE  PRISM. 

82.  In  general,  as  we  have  seen  (§81),  the  astigmatic  difference 
of  the  bundle  of  emergent  rays  arising  from  the  refraction  through  a 
prism  of  an  infinitely  narrow  homocentric  bundle  of  incident  rays, 
having  its  chief  ray  in  a  principal  section  of  the  prism,  will  not  be  zero; 
that  is,  the  I.  and  II.  Image-Points  S2,  ^  will  not  coincide  in  one  point 
So  on  the  chief  emergent  ray  u'2.  If  this  does  happen,  then  the  image 
of  a  point-source  Sl  as  seen  through  a  prism  will  be  a  point  £,.  In 
certain  cases,  as  has  been  very  beautifully  shown  by  Prof.  Dr.  L. 

1  This  law  in  its  general  form,  as  here  stated,  is  given  by  L.  BURMESTER:  Homocentrische 
Brechung  des  Lichtes  durch  das  Prisma:  Zft.  f.  Math.  u.  Phys.,  xl.  (1895),  65-90.  For 
the  case  when  the  prism  is  surrounded  by  the  same  medium  on  both  sides,  the  law  was  ob- 
tained also  by  A.  GLEICHEN:  Ueber  die  Brechung  des  Lichtes  durch  Prismen:  Zft.f.  Math, 
u.  Phys.,  xxxiv.  (1889),  161-176. 

8 


98  Geometrical  Optics,  Chapter  IV.  [  §  83. 

BuRMESTER,1  this  can  occur;  although  until  the  publication  of  BUR- 
HESTER'S  investigations  on  this  subject,  the  laws  of  the  homocentric 
refraction  of  light  through  a  prism  appear  not  to  have  been  clearly 
formulated  except  for  certain  special  cases.2 

In  the  following  discussion  we  shall  show  how  the  main  results 
obtained  by  BURMESTER  by  purely  geometrical  methods  may  be  de- 
duced from  the  general  formula  (34)  for  the  astigmatic  difference,  as 
is  done  by  LoEWE;3  and  we  shall  give  also  an  outline  of  the  elegant 
geometrical  method  used  by  BURMESTER  himself. 

83.  Analytical  Method.  In  the  first  place  we  may  remark  that 
when  the  incident  rays  are  an  infinitely  narrow  bundle  of  parallel  rays 
(st  =  oo),  the  ratio  (s'2  —  s'^/Sj^  will  be  vanishingly  small;  that  is, 
the  astigmatic  difference  will  be  practically  equal  to  zero  in  com- 
parison with  the  distance  from  the  prism  of  the  point-source  of  light. 
This  is  the  essential  advantage  of  using  parallel  incident  rays  in  work- 
ing with  a  prism-spectroscope. 

The  condition  that  the  astigmatic  difference  1>2S2  °f  the  bundle  of 
emergent  rays  shall  vanish,  that  is,  that  the  I.  and  II.  Image-Points 
on  the  emergent  chief  ray  u'2  shall  coincide  in  a  single  point  22,  is 
found  immediately  by  putting  s'2  —  s'2  =  S2S2  =  o  in  formula  (34),  and 
is  as  follows: 


2  2  2  -  '  2 

COS    «!  •  COS    OL2  —  COS    ^  •  COS 


where  2^  designates  the  homocentric  Object-Point  on  the  incident  chief 
ray  uv  to  which  corresponds  the  homocentric  Image-Point  22  on  the 
emergent  chief  ray  u'z.  This  distance  B1'2l  is  determined  by  this  equa- 
tion as  a  unique  function  of  ^  and  5X;  the  two  magnitudes  which, 
for  a  given  prism,  define  completely  the  incident  chief  ray  u^  Hence, 
equation  (35)  shows  that: 

On  every  incident  chief  ray  u^  refracted  through  a  prism  in  a  principal 
section,  there  is  in  general  one,  and  only  one,  Object-  Point  Sx  to  which 
on  the  emergent  chief  ray  u'2  there  corresponds  a  homocentric  Image- 
Point  2'2. 

Moreover,  for  a  given  value  of  the  angle  of  incidence  alf  the  length 

'L.  BURMESTER:  Homocentrische  Brechung  des  Lichtes  durch  das  Prisma:  Zft.  f. 
Math.  u.  Phys.,  ad.  (1895),  65-90. 

2  See  H.  HELMHOLTZ:  Wissenschaftliche  Abhandlungen,  Bd.  II  (Leipzig,  1883),  S.  167. 
A.  GLEICHEN:  Ueber  die  Brechung  des  Lichtes  durch  Prismen:  Zft.f.  Math.  u.Phys.,  xxxiv. 
(1889),  161-176.  J.  WILSING:  Zur  homocentrische  Brechung  des  Lichtes  durch  das 
Prisma:  Zft.  f.  Math.  u.  Phys.,  xl.  (1895),  353-361. 

9F.  LOEWE:  Die  Prismen  und  die  Prismensysteme:  Chapter  VIII  of  Die  Theorie  der 
optischen  Inslrumente,  Bd.  I  (Berlin,  1904),  herausgegeben  von  M.  VON  ROHR.  See  p.  433. 


§  84.]  Refraction  Through  a  Prism  or  Prism-System.  99 

of  the  ray-path  within  the  prism,  denoted  by  5,,  is  proportional  to 
the  distance  VBl  of  the  incidence-point  from  the  refracting  edge  of 
the  prism,  and  if  Bl  coincides  with  F,  5L  =  o;  and,  therefore: 

Object-  Points  lying  upon  parallel  incident  chief  rays  which  are  re- 
fracted through  the  prism  in  a  principal  section,  which  have  homocentric 
Image-  Points,  are  all  contained  in  a  certain  plane  which  passes  through 
the  edge  of  the  prism. 

Thus,  a  bundle  of  parallel  incident  chief  rays  determines  a  remark- 
able plane  passing  through  the  edge  of  the  prism  which  is  character- 
ized by  the  property  that  each  Object-  Point  in  this  plane  is  imaged  in 
the  prism  by  a  homocentric  Image-Point.  For  another  bundle  of 
parallel  incident  chief  rays  we  shall  have  another  such  plane  passing 
through  the  refracting  edge  of  the  prism  and  characterized  in  the  same 
way.  Moreover,  it  is  easy  to  show  similarly  that  the  locus  of  the 
homocentric  image-points  for  such  a  system  of  parallel  incident  chief 
rays  is  also  a  plane  passing  through  the  refracting  edge  of  the  prism. 
The  angles  made  with  the  faces  of  the  prism  by  each  of  these  two 
corresponding  planes,  one  of  which  is  the  locus  of  the  Object-  Points 
and  the  other  the  locus  of  the  corresponding  homocentric  Image-Points, 
for  a  given  direction  ^  of  the  parallel  incident  chief  rays,  can  also 
be  easily  calculated,  as  is  done  by  LOEWE. 

A  rather  exceptional  case,  however,  is  the  case  when  the  incident 
chief  ray  has  a  direction  such  that 

cos  o^  •  cos  a2  =  cos  a[  •  cos  ct2  ; 

for  then  the  denominator  of  the  fraction  on  the  right-hand  side  of 
equation  (35)  vanishes,  and  we  have  therefore  .B^  =  °o  ;  in  which 
case  we  have  also  B^  =  oo  ;  so  that  for  this  particular  direction  of 
the  chief  incident  ray  both  the  Object-Point  and  the  corresponding 
homocentric  Image-Point  lie  at  infinity.  If  the  prism  is  surrounded 
on  both  sides  by  the  same  medium,  the  equation  above  is  the  condi- 
tion that  the  chief  ray  shall  traverse  the  prism  with  minimum  devia- 
tion (§  71,  §  81). 

84.  Geometrical  Investigation  (according  to  Burmester).  In  a 
principal  section  of  the  prism  let  us  consider  a  system  of  parallel  inci- 
dent chief  rays  of  which,  for  example,  u^  is  one  ray.  Corresponding 
to  a  range  of  Object-Points 


on  the  incident  chief  ray  ult  we  have,  according  to  §  63,  a  similar  range 


100  Geometrical  Optics,  Chapter 

of  I.  Image-Points 

PU.II     Qu.li    RU.II     '  '  ' 

and  a  similar  range  of  II.  Image-Points 


both  lying  on  the  chief  ray  u{  of  the  bundle  of  rays  refracted  at  the 
first  face  of  the  prism.  And  so  corresponding  to  a  range  of  Object- 
Points  on  each  incident  chief  ray  of  the  system  of  parallel  rays  we  shall 
have  two  similar  ranges  lying  on  the  corresponding  chief  ray  of  the 
bundle  of  rays  refracted  at  the  first  face  of  the  prism.  This  system 
of  Object-Points  lying  on  parallel  incident  chief  rays,  such  as  ultvlt  etc., 
may  be  referred  to  as  a  whole  as  the  system  ^  ;  and  to  this  system  of 
Object-Points  ^  there  corresponds,  as  explained  in  §63,  a  system  of 
I.  Image-Points  r)(  and  a  system  of  II.  Image-Points  77"!  lying  on  the 
parallel  chief  rays  of  the  bundles  of  rays  which  are  refracted  at  the 
first  face  of  the  prism.  Each  of  these  systems  rj[  and  771  is  in  affinity 
with  the  system  of  Object-Points  77^ 

Again,  corresponding  to  the  system  ??{,  we  have  a  system  of  I.  Image- 
Points  772  which  lie  on  the  rays  of  the  pencil  of  parallel  emergent  chief 
rays,  which  is  likewise  in  affinity  with  the  system  rj[.  And,  similarly, 
corresponding  to  the  system  rj[,  we  have  a  system  of  II.  Image-Points 
^2  which  lie  on  the  rays  of  the  pencil  of  parallel  emergent  chief  rays, 
which  is  likewise  in  affinity  with  the  system  rj[. 

Since  the  system  of  Object-  Points  TJI  is  in  affinity  with  the  systems 
»h  and  r}[,  and  since  T/[  is  in  affinity  with  rj'2,  and  r/l  is  in  affinity  with  TJ'2, 
it  follows  that  the  system  ^  is  in  affinity  with  both  ^  and  7J'2;  and, 
hence,  also  the  systems  772,  rj2  are  in  affinity  with  each  other. 

The  three  systems  77^  77  [  and  77  J,  which  are  in  affinity  each  with  the 
other,  have  a  common  affinity-axis,  viz.,  the  straight  line  VBl  in  which 
the  plane  of  the  principal  section  meets  the  first  face  of  the  prism. 

The  straight  line  VB2  in  which  the  plane  of  the  principal  section 
meets  the  second  face  of  the  prism  is  the  affinity-axis  of  the  two  sys- 
tems 77^  and  T?^;  and  this  straight  line  is  also  the  affinity-axis  of  the 
two  systems  V^  and  rj2.  The  point  V  in  which  the  plane  of  the  prin- 
cipal section  meets  the  refracting  edge  of  the  prism  is  on  both  of  these 
affinity-axes;  so  that  for  each  pair  of  the  five  systems  77^  iy[,  771,  772,  772 
which  are  in  affinity  with  each  other  the  point  V  is  a  self  -cor  responding 
point. 

The  three  points  5,,  S{  and  S2  on  the  corresponding  chief  rays  wt 
u{  and  u'2  (Fig.  41),  or  the  three  points  Zlf  Z(  and  Z2  on  the  correspond- 


§84.] 


Refraction  Through  a  Prism  or  Prism-System. 


101 


ing  chief  rays  zlt  z(  and  z'2  parallel  to  ult  u[  and  u'2,  respectively,  deter- 
mine the  three  systems  tj^  r}{  and  rj'2  which  are  in  affinity  each  with 
the  other. 

N 


FIG.  42. 

HOMOCENTRIC  REFRACTION,  THROUGH  A  PRISM,  OF  INFINITELY  NARROW  BUNDLE  OF  RAYS( 
WITH  CHIEF  RAY  (as  u\)  LYING  IN  A  PRINCIPAL  SECTION  OF  PRISM.  To  the  object-point  2i  on  the 
chief  incident  ray  u\  corresponds  the  homocentric  image-point  2a'  on  the  chief  emergent  ray  uz'- 

In  the  same  way,  the  three  corresponding  chief  rays  ult  u(  and  u'2 
or  zly  z(  and  z'2  are  sufficient  to  determine  the  three  systems  rjv  ^  and 


102  Geometrical  Optics,  Chapter  IV.  [  §  84. 

TJ.',  which  are  in  affinity  each  with  the  other;  for  the  corresponding 
points  of  ijlf  y(  lie  on  the  normals  to  the  first  face  of  the  prism,  and  the 
corresponding  points  of  ?|,  rj'2  lie  on  the  normals  to  the  second  face  of 
the  prism. 

Since  the  corresponding  points  of  the  two  systems  rj2J  rj'2  lie  on  the 
rays  of  the  pencil  of  parallel  emergent  chief  rays,  the  affinity-axis  of 
this  pair  of  systems  must  go  through  the  double-point  F;  and  along 
this  straight  line,  which  we  shall  denote  by  a'2,  must  lie  the  self-cor- 
responding points  of  r)2  and  rj2,  or  the  Homocentric  Image- Points  of 
the  pencil  of  parallel  emergent  chief  rays. 

This  affinity-axis  may  be  constructed  by  determining  the  point  of 
intersection  of  two  corresponding  straight  lines  of  the  systems  rj2  and 
rj'2.  In  the  figure  (Fig.  42)  the  points  Zlf  Z2  lying  on  the  incident 
and  emergent  chief  rays  zlt  z'2,  respectively,  to  which  on  the  chief  ray 
z{  of  the  bundle  of  rays  refracted  at  the  first  face  of  the  prism  corre- 
sponds the  I.  Image-Point  Zj,  are  constructed  exactly  as  was  described 
in  §  76  (see  Fig.  41);  as  are  also  the  two  pairs  of  I.  and  II.  Image- 
Points  S{,  5[  and  S2,  32  corresponding  to  the  object-point  Sl  on  the 
chief  incident  ray  ^  drawn  parallel  to  the  chief  incident  ray  %,  which 
latter  ray  meets  the  refracting  edge  of  the  prism. 

To  the  point  B±  where  the  chief  incident  ray  u\  meets  the  first  face 
of  the  prism,  regarded  as  an  object-point  on  this  ray,  there  correspond 
on  the  emergent  chief  ray  u'2  (as  was  also  explained  in  §  81)  the  I.  and 
II.  Image-Points  B2,  B2,  which  are  constructed  by  drawing  from  B1 
two  straight  lines,  one  parallel  to  the  straight  line  Z^Z^  and  meeting 
u'2  in  B2,  and  the  other  perpendicular  to  the  second  face  of  the  prism 
and  meeting  u2  in  B2. 

The  II.  Image-Point  Z2  corresponding  to  the  Object-Point  Zt  on 
the  chief  incident  ray  z±  may  be  found  by  drawing  from  Zl  a  straight 
line  perpendicular  to  the  first  face  of  the  prism  meeting  the  ray  z\ 
in  the  point  Zj,  and  from  Zj  a  straight  line  perpendicular  to  the  second 
face  of  the  prism  meeting  the  corresponding  chief  emergent  ray  z'2 
in  Z2.  The  points  B2,  B2  and  Z2J  E2,  are  two  pairs  of  corresponding 
points  of  the  systems  772,  rj'2',  and  hence  B2Z2J  B'2Z2  are  a  pair  of  cor- 
responding straight  lines  of  these  systems ;  which  must  therefore  inter- 
sect in  a  point  &2  lying  on  the  affinity-axis  a'2  of  the  two  systems  rj2,  rj'2. 
Accordingly,  the  affinity-axis  a'2  is  the  straight  line  Vtt'2.  Instead  of 
using  here  the  pair  of  corresponding  points  B2,  B2,  we  may  use  also 
any  other  pair  as  S2,  S2  on  u'2,  in  conjunction  with  the  pair  Z2J  %2  on 
z'2,  which  will  determine  some  other  point  SI^  on  tne  affinity-axis  a'2. 

The  point  22  where  the  emergent  ray  u'2  meets  the  affinity-axis  a'2 


§  84.]  Refraction  Through  a  Prism  or  Prism-System.  103 

is  the  double-point,  or  homocentric  Image-  Point,  of  the  two  similar 
ranges  of  I.  and  II.  Image-Points  lying  on  the  emergent  chief  ray  u'2 
which  correspond  to  a  similar  range  of  Object-Points  lying  on  the  in- 
cident chief  ray  u^ 

This  point  2'2  may  also  be  constructed  by  producing  S[S2  and  S'jS2 
to  meet,  say,  in  a  point  W\  then  the  point  where  the  straight  line  WBl 
meets  the  emergent  chief  ray  u'2  will  be  the  double-point  22  of  the  two 
similar  ranges  of  I.  Image-Points  (S2,  B'2,  -  •  •)  and  II.  Image-Points 
(S2  B2,  •.••')  lying  along  the  emergent  chief  ray  u2.  For,  evidently, 
according  to  this  construction,  we  have: 


s2s2     2;  w 

and  since  the  point  ranges  S2,  B2,  •  •  •  and  S2J  B2J  •  •  •  are  similar, 
obviously  the  point  £2  must  be  the  self-corresponding  or  double-point 
of  these  two  ranges  of  points  lying  together  on  the  emergent  chief 
ray  u'2. 

The  Object-Point  Sx  on  the  incident  chief  ray  ult  to  which  on  the 
emergent  chief  ray  u'2  corresponds  the  homocentric  Image-Point  22 
may  be  constructed  in  either  of  two  ways  as  follows:  From  the  point 
2  2  draw  a  straight  line  parallel  to  Z2Z(  meeting  u(  in  the  point  1<{\ 
and  from  this  point  ^(  draw  a  straight  line  parallel  to  Z(Z±  which 
will  determine  by  its  intersection  with  u±  the  required  Object-  Point 
2^  Or,  from  S2  draw  a  straight  line  perpendicular  to  the  second  face 
of  the  prism  meeting  u(  in  the  point  S[  ;  and  from  this  point  Sj  draw 
a  straight  line  perpendicular  to  the  first  face  of  the  prism  which  will 
likewise  meet  the  incident  chief  ray  u^  in  the  required  point  2X. 

These  Object-  Points  S^  Flf  M^,  0L,  •  •  •  of  the  pencil  of  parallel  inci- 
dent chief  rays  to  which  correspond  the  homocentric  image-points  S2, 
V,  ^2,  Q'2,  v-  •  all  lying,  as  we  saw,  on  the  affinity-axis  a'2  of  the  systems 
1/2,  rj'2,  will  themselves  also  lie  on  a  straight  line  at  meeting  a'2  in  the 
point  F,  the  two  straight  lines  at  and  a'2  corresponding  to  each  other 
as  incident  and  emergent  rays. 

From  the  foregoing  results  it  follows  that  the  point  Sx  is  the  only 
Object-Point  on  the  incident  chief  ray  u^  to  which  on  the  emergent 
chief  ray  u'2  there  corresponds  a  homocentric  Image-Point  2^;  and, 
hence,  on  every  such  chief  ray  refracted  through  a  prism  in  a  principal 
section  there  is  always  one,  and  only  one,  pair  of  points  which  are, 
so  to  speak,  in  Homocentric  Correspondence  with  each  other;  exactly 
as  we  saw  also  in  §  83.  Moreover,  Object-Points,  lying  on  the  rays  of 
a  pencil  of  parallel  incident  chief  rays  in  a  principal  section  of  the 


104  Geometrical  Optics,  Chapter  IV.  [  §  85. 

prism,  to  which  on  the  rays  of  the  pencil  of  parallel  emergent  rays 
correspond  Homocentric  Image-Points,  are  ranged  along  a  straight  line 
Fat  which  goes  through  the  refracting  edge  of  the  prism;  and  the 
corresponding  Homocentric  Image-Points  are  ranged  likewise  along 
a  straight  line  Va'2  which  may  be  regarded  as  the  emergent  ray  cor- 
responding to  the  incident  ray  Va^  All  these  results  are  in  agree- 
ment with  those  found  in  §  83.  These  laws  were  distinctly  formulated 
first  by  BURMESTER,  and  for  a  further  account  of  his  investigations 
the  reader  is  referred  to  his  original  paper  on  this  subject. 

85.  In  order  to  verify  his  results,  BURMESTER  employed  a  glass 
prism  of  refracting  angle  60°,  for  which  the  value  of  n  for  the  FRAUN- 
HOFER  D-line  was  n  =1.7.  The  prism  is  shown  in  section  in  the 
diagram  (Fig.  43)  which  gives  the  disposition  of  the  apparatus.  On 


FIG.  43. 
SHOWING  THE  PLAN  OF  BURMESTER'S  EXPERIMENT. 

a  certain  incident  ray  ulBl  the  Object-Point  Si  to  which  corresponds 
a  homocentric  Image-Point  22  on  the  emergent  chief  ray  u2  was  con- 
structed; and,  moreover,  the  I.  and  II.  Image-Points  S'2  and  S'2  cor- 
responding to  an  arbitrary  Object-Point  Sl  on  ^  were  also  constructed 
by  the  methods  given  above.  The  prism  was  supported  on  a  block, 
which  was  movable  in  parallel  guides  in  a  direction  parallel  to  the 
straight  line  u2.  On  this  same  block  was  placed  a  glass  cube  with 
two  of  its  parallel  faces  perpendicular  to  the  incident  chief  ray  ulf 
the  face  nearest  to  the  prism  being  at  the  distance  Bl2l  from  it.  This 
face  was  covered  with  lamp-black  except  at  the  point  St  where  a  small 
opening  was  made  with  a  needle.  A  sodium-flame  F  was  placed  on 
the  block  with  the  prism  and  the  glass  cube.  Finally,  an  ABBE'S 
Focometer  A,  for  which  the  distance  from  the  objective  0  of  a  dis- 
tinctly visible  object  is  equal  to  no  mm.,  was  placed  in  a  fixed  posi- 
tion with  its  axis  coinciding  with  the  emergent  chief  ray  u'2,  and  the 


§  86.]  Refraction  Through  a  Prism  or  Prism-System.  105 

movable  block  was  displaced  with  respect  to  the  fixed  focometer  so 
that  the  distance  22O  =110  mm.  These  adjustments  having  been 
completed,  the  homocentric  Image-Point  at  22  resulting  from  the  re- 
fraction through  the  prism  of  the  narrow  homocentric  bundle  of  inci- 
dent rays  proceeding  from  the  point-source  at  2^  was  seen  through 
the  focometer  as  a  "small  bright  opening  so  distinctly  that  even  the 
roughness  of  the  contour  of  the  hole  in  the  layer  of  lamp-black  could 
be  clearly  recognized." 

Moreover,  when  the  glass  cube  was  placed  on  the  movable  block 
with  its  blackened  face  at  the  distance  BlSl  from  the  prism,  so  that 
the  pin-hole  opening  in  the  layer  of  lamp-black  was  at  the  Object- 
Point  Slt  and  when  the  block  was  displaced  with  respect  to  the  fixed 
focometer  so  that  the  distance  S'20  =  no  mm.,  the  image  of  the  point- 
source  Sj  as  seen  through  the  focometer  was  a  small  straight  line  par- 
allel to  the  (vertkjal)  edge  of  the  prism;  whereas  if  the  block  was 
displaced  so  that  S'20  =  no  mm.,  the  image  was  a  small  horizontal 
line. 

ART.  26.     APPARENT    SIZE    OF    IMAGE    OF    ILLUMINATED    SLIT    AS    SEEN 

THROUGH  A  PRISM. 

86.  If  the  source  of  the  incident  rays  of  light  is  a  narrow  illumi- 
nated slit,  with  its  length  parallel  to  the  prism-edge,  as  is  the  usual 
arrangement  in  the  prism-spectroscope,  and  if  we  adjust  the  eye  (or 
telescope)  to  view  the  image  of  the  slit  formed  by  the  sagittal  rays, 
a  clear  and  distinct  image,  practically  unaffected  by  astigmatism,  will 
be  seen,  as  was  mentioned  in  §  79. 

If  the  apparent  breadth  and  the  apparent  height  of  the  slit  as  seen 
from  the  point  of  incidence  #L  at  the  first  face  of  the  prism  are  denoted 
by  db  and  dh]  and  if  the  apparent  breadth  and  the  apparent  height 
of  the  II.  Image  of  the  slit  as  seen  from  the  point  of  emergence  B2 
at  the  second  face  of  the  prism  are  denoted  by  db'  and  dhf,  then, 
evidently : 

dV-r    -^2       dh/_  _-        d\2. 
db~Zu~  da,'      dh    "    Z»~  d\' 

where  the  magnitudes  denoted  by  Zu,  ~ZU  are  determined  by  formulae 
(32)  and  (33).  If  the  prism  is  surrounded  by  the  same  medium  on 
both  sides,  so  that  n'2  =  n,  then  Zu  =  i ;  so  that  in  this  case  the  ap- 
parent height  of  the  slit-image  is  equal  to  the  apparent  height  of  the 
slit  itself. 

But  the  apparent  breadth  of  the  slit-image  will,  in  general,  be  differ- 


106  Geometrical  Optics,  Chapter  IV.  [  §  87. 

ent  from  that  of  the  slit;  for,  since 

cos<vcos<*2          , 

ZM    =    -  '»         (H2    —   ni)l 

cos  <*!  •  cos  «2 

it  appears  that  the  value  of  ZM  will  depend  on  the  angle  of  incidence 
«!.  If,  for  example,  either  a(  or  a'2  =  90°,  the  value  of  Zu  will  be 
infinite,  and  hence  dbf  =  oo.  On  the  other  hand,  if  one  of  the  angles 
in  the  numerator,  for  example,  a^  =  90°  (case  of  so-called  "grazing 
incidence"),  we  have  Zu  =  o  and,  therefore,  also  db'  =  o.  Thus,  the 
image  of  the  slit  may  appear  infinitely  broad  or  infinitely  narrow,  and 
may  have  any  apparent  breadth  between  these  two  extremes  depend- 
ing on  the  value  of  the  angle  of  incidence  ar 

When  the  rays  proceed  through  the  prism  with  minimum  deviation 
(n'2  =  »!,  cos  <%! -cos  a2  =  cos  aj-cos  a'2) ,  we  have  Zu  =  I ;  so  that  then 
both  the  apparent  height  and  breadth  of  the  slit-image  are  equal  to 
the  apparent  height  and  breadth  of  the  slit  itself. 

ART.  27.     ASTIGMATIC  REFRACTION  OF  INFINITELY  NARROW,  HOMO  CEN- 
TRIC BUNDLE  OF  INCIDENT  RAYS  ACROSS  A  SLAB  WITH 
PLANE  PARALLEL  FACES. 

87.  As  we  have  seen,  a  Slab  or  Plate,  with  parallel  plane  refracting 
faces,  may  be  treated  as  a  prism  whose  refracting  angle  is  equal  to 
zero;  so  that  the  methods  and  formulae  of  the  preceding  articles  can 
be  adapted  to  this  problem  by  treating  the  slab  as  a  special  case  of 
the  prism.  For  the  sake  of  generality,  let  us  assume  that  the  media  of 
the  incident  and  emergent  rays  are  different;  and  let  us  denote  the 
absolute  indices  of  refraction  of  the  three  media,  in  the  order  in  which 
they  are  traversed  by  the  rays  of  light,  by  »lf  n{  and  n'2.  We  shall 
give,  first,  the  construction  of  the  I.  and  II.  Image- Points  correspond- 
ing to  an  Object- Point  on  a  given  incident  chief  ray  u\. 

In  the  figure  (Fig.  44)  the  plane  of  the  paper  represents  the  plane 
of  incidence  of  the  incident  chief  ray  u^  which  meets  the  first  face  of 
the  slab  at  the  point  Blf  With  .Bl  as  centre  and  with  radii  equal  to 
r,  nflnd  and  n^rjn^  (where  r  denotes  any  arbitrary  length)  describe 
the  arcs  of  three  concentric  circles;  and  through  the  point  G  where 
the  circle  of  radius  n^jn^  meets  the  incident  chief  ray  «,,  draw  a 
straight  line  normal  to  the  first  face  of  the  slab,  and  let  this  straight 
line  meet  the  circles  of  radii  r  and  n'2r/n[  in  the  points  S(  and  /,  respec- 
tively. Then,  exactly  as  in  §  74,  the  straight  line  S[Bl  will  determine 
the  path  of  the  ray  B1B2  or  u[  after  refraction  at  the  first  face  of 
the  slab,  and  the  path  of  the  emergent  ray  u2  is  determined  by  draw- 
ing through  B2  a  straight  line  parallel  to 


88.] 


Refraction  Through  a  Prism  or  Prism-System. 


107 


Through  S(  draw  a  straight  line  perpendicular  to  S^  meeting  the 
first  face  of  the  slab  in  the  point  £/,.  From  U^  draw  the  straight 
lines  lift  and  U^J  meet- 
ing at  X  and  F,  respec- 
tively, the  incidence- 
normal  B^NV  and  from 
X  and  F  let  fall  on  GB, 
and  JBl  the  perpendicu- 
lars XSt  and  YZ,  re-  W 
spectively;  and,  finally, 
draw  S(Z  meeting  the 
emergent  chief  ray  u'2  in 
the  point  S'2.  Then  S2  is 
the  I.  Image-Point  on 
the  chief  emergent  ray 
u'2  corresponding  to  the 
Object- Point  Sl  on  the 
chief  incident  ray  «L ;  as 
is  evident,  since  the  con- 
struction given  here  is 
merely  a  special  case  of 
the  construction  in  the 
case  of  a  prism  (§  76). 

The  II.  Image-Point 
S'2  corresponding  to  the 
Object- Point  Sl  on  the 
chief  incident  ray  u^  is 
found  by  drawing 
through  Sl  a  straight  line  perpendicular  to  the  first  face  of  the  slab, 
which  will  meet  the  emergent  chief  ray  u2  in  the  required  point. 

88.  Formulae  for  the  Determination  of  the  Positions  of  the  I.  and 
II.  Image-Points.  Since  in  the  case  of  a  slab  with  plane  parallel  faces 
we  have  a{  =  a2,  we  have  merely  to  introduce  this  condition  in  the 
formulae  (30)  and  (31)  in  order  to  obtain  the  corresponding  formulae 
for  this  special  case.  Thus,  employing  also  the  relations: 


FIG.  44. 

REFRACTION  OF  NARROW  BUNDLE  OF  RAYS  ACROSS  A 
SLAB  WITH  PLANE  PARALLEL  FACES.  Construction  of  the 
homocentric  image-point  on  the  chief  emergent  ray  corre- 
sponding to  a  given  chief  incident  ray. 


sn 


=  n-  sn 


n-  sin  a  =  n-  sn 


we  derive  easily  the  following  relations  for  the  determination  of  the 
positions  of  the  Image-Points  in  the  case  of  an  infinitely  narrow,  homo- 
centric  bundle  of  incident  rays  refracted  across  a  slab  with  plane 
parallel  faces: 


108  Geometrical  Optics,  Chapter  IV.  [  §  89. 

Meridian  Rays: 

'     -co'      n^-nj-sin'o^  f        sl  nfo  1 

*2  =  *A  =        —  >-     -  {-  cos2  ai  -  w,«  _  w?>  sin2  J  ,  (36) 

Sagittal  Rays: 

(37) 


In  these  formulae,  ^  =  -BA,  5i  =  B\B* 

Similarly,  by  specializing  formulas  (32)  and  (33),  we  obtain  for  the 
Convergence-  Ratios  of  the  meridian  and  sagittal  rays  in  the  case  of  a 
slab  with  plane  parallel  faces: 
Meridian  Rays: 

da'2       M!  cos  <*!  . 

Z,.  =    ~;  —  =  —  7  ~         /»  v3°7 

dc*!        W2  COS  «2 

Sagittal  Rays: 

d\y  Wi 

z»=^;=4-          _  (39) 

In  the  special  case  when  we  have  the  same  medium  on  both  sides  of 
the  slab,  the  formulae  above  may  be  simplified  by  putting  n  =  n^jn^ 
—  n(ln2,  in  which  case,  in  addition  to  the  condition  a[  =  «2>  we  have 
also  «o  =  «i«  Thus,  we  obtain: 


COS 
Sl 2" 

1      cos 


i. 


(40) 


In  case  the  slab  is  at  the  same  time  very  thin,  so  that  B1B2  is  practi- 
cally negligible,  we  have  approximately  SL  =  s'2  =  ~s'2. 


89.  Astigmatic  Difference  in  Case  of  a  Slab.  The  formula  for  the 
astigmatic  difference  of  the  bundle  of  emergent  rays  corresponding  to 
an  infinitely  narrow  homocentric  bundle  of  incident  rays  refracted 
across  a  slab  with  plane  parallel  faces  may  be  obtained  by  combining 
formulae  (36)  and  (37),  or,  perhaps  more  simply  still,  by  introducing 
in  formula  (34)  the  condition  a[  =  a2\  thus,  we  obtain: 

-«/  c/        /      -/      n'2  /cos2  a'2         \         n2  /cos2  a'2        \ 

S2S2  =  s2  -  s2  =  —  \  — 2—^  -  i  ]sl 7 1      -,—  -  i   6,.        (41) 

nl  \cos  «!         /         »i  \cos  a2        )  l 

In  general,  therefore,  the  astigmatic  difference  for  a  given  chief  inci- 
dent ray  will  depend  on  the  position  on  this  ray  of  the  radiant  point  5X. 


§  89.]  Refraction  Through  a  Prism  or  Prism-System.  109 

The  condition  that  the  astigmatic  difference  of  the  bundle  of  emer- 
gent rays  shall  be  independent  of  the  ray-distance  (sj  of  the  Object- 
Point  5i  from  the  incidence-point  Bl  is  evidently: 

cos2  a'2 

2       -  i  =  o; 
cos  aL 

which  implies  here  that  we  have  a'2  =  a^  and  this  in  turn  involves 
either  (i)  that  we  have  the  same  medium  on  both  sides  of  the  slab 
(n'2  =  nj,  or  (2)  that  the  chief  ray  u^  is  incident  normally  on  the  slab, 
so  that  «2  =  «!  =  o.  We  consider  briefly  each  of  these  cases. 

(i)  Slab  surrounded  by  same  medium  on  both  sides.  We  obtain  in 
this  case: 


where  n  =  n^/^  =  n[/n'2. 

(2)  Normal  Incidence  (c^  =  a(  =  a2  =  a'2  =  o).  In  this  case  the 
astigmatic  difference  vanishes,  and  the  bundle  of  emergent  rays  is 
homocentric.  To  every  object-point  on  a  normally  incident  chief  ray 
there  corresponds  a  homocentric  Image-Point.  If  on  the  normally 
incident  chief  ray  N1B1  (Fig.  44)  we  take  any  Object-  Point  Ml  (not 
marked  in  the  figure)  to  which  on  the  corresponding  normally  emergent 
chief  ray  there  corresponds  the  homocentric  image-point  M2,  then,  by 
formula  (36)  or  formula  (37),  we  obtain: 


where  dl  denotes  the  thickness  of  the  slab  (dl  =  -B^);  which  may 
also  be  put  in  the  following  form: 


If  B1M1  =  BiM'2  =  B£),  so  that  the  homocentric  Image-Point  co- 
incides at  the  point  0  with  the  Object-Point  Mlt  this  double-point  0 
of  the  two  similar  ranges  of  Object-  Points  and  Image-Points  lying  on 
a  straight  line  perpendicular  to  the  faces  of  the  slab  can  be  located  by 
the  following  formula  : 


Hence,  on  a  normally  incident  chief  ray  refracted  across  a  slab  with 
plane  parallel  faces  there  can  always  be  found  a  certain  point  0  at 


110  Geometrical  Optics,  Chapter  IV.  [  §  90. 

which  the  Object-Point  and  its  Homocentric  Image-Point  coincide  with 
each  other.  If  the  slab  is  surrounded  by  the  same  medium  on  both 
sides,  this  point  0  lies  at  infinity. 

When  a  luminous  point  Ml  is  viewed  normally  through  a  trans- 
parent slab  with  parallel  plane  faces,  the  displacement  in  the  line  of 
vision  of  the  Homocentric  Image-Point  M2  with  respect  to  the  Object- 
Point  Ml  is: 


-  i    B.M,  + 

and  if  the  slab  is  surrounded  by  the  same  medium  on  both  sides 
(n  =  »!/»!  =  n't/n't),  we  obtain: 


a  formula  which,  according  to  a  method  suggested  by  Due  DE  CHAUL- 
NES  (1767),  is  employed  for  the  determination  of  the  relative  index  of 
refraction  (n),  the  lengths  M^M^  and  d^  being  both  capable  of  easy 
measurement. 

90.  Exactly  as  in  the  case  of  refraction  through  a  prism  (§  84),  we 
can  construct  on  every  incident  chief  ray  u^  of  a  narrow  bundle  of 
incident  rays  refracted  across  a  slab  with  parallel  plane  faces  the 
Object-Point  St  to  which  on  the  chief  ray  u2  of  the  bundle  of  emergent 
rays  there  corresponds  the  Homocentric  Image-Point  "22.  Thus,  draw- 
ing through  Sl  (Fig.  44)  a  straight  line  perpendicular  to  the  first  face 
of  the  slab  and  meeting  the  straight  line  S[S2  in  the  point  W,  we  find 
the  Homocentric  Image-Point  22  at  the  point  of  intersection  of  B^W 
with  the  emergent  chief  ray  Uy  A  straight  line  drawn  through  S2 
perpendicular  to  the  first  face  of  the  slab  will  determine  by  its  inter- 
section with  the  incident  chief  ray  u^  the  Object-Point  SL  which  cor- 
responds to  the  Homocentric  Image-Point  "22. 

The  formula  for  the  determination  of  the  position  on  a  given  inci- 
dent chief  ray  u^  of  the  Object-  Point  St  which  has  a  Homocentric 
Image-Point  can  be  obtained  from  formula  (41)  by  equating  to  zero 
the  right-hand  side  of  this  equation.  Thus,  writing  here  B^  in  place 
of  slt  and  also  employing  the  relations: 

n^  -  sin  oij  =  n{  •  sin  a{  ,     n[  -  sin  «2  =  n'2  -  sin  a'2, 
we  obtain: 

3  /2  /2  n 

Ry   _ni  "2  -*i   cos   a,     - 

•^l^l  ~    ~3  ~72  --  j  ---  2  -  '  51» 

n{  n2  —  n\   cos  at 


§  91.]  Refraction  Through  a  Prism  or  Prism-System.  HI 

whence  it  is  seen  that  the  distance  of  the  Object-Point  Sx  from  the 
incidence-point  BL  is  proportional  to  the  length  of  the  ray-path  5X 
within  the  slab,  that  is,  is  proportional  to  the  thickness  of  the  slab, 
since  ^  =  S^cosa,'.  BURMESTER/  employs  this  formula  to  obtain  a 
very  simple  geometrical  construction  of  the  Object-Point  Sj. 

In  the  case  of  normal  incidence,  where  the  angles  of  incidence  and 
refraction  at  both  faces  of  the  slab  are  equal  to  zero,  the  value  of  ^ 
as  given  by  formula  (41)  is  indeterminate;  that  is,  to  every  Object- 
Point  on  a  normally  incident  chief  ray  there  corresponds  a  Homocen- 
tric  Image-Point;  as  we  saw  also  above. 

If  the  slab  is  surrounded  by  the  same  medium  on  both  sides,  we 
find  from  the  formula  just  derived  that  B^  =  oo  for  all  angles  of 
incidence.  In  this  special  case,  therefore,  the  Homocentric  Image- 
Points  and  the  Object-Points  to  which  they  correspond  are  both  at 
an  infinite  distance. 

ART.  28.     PATH  OF  A  RAY  REFRACTED  THROUGH  A  SYSTEM  OF  PRISMS,  IN 

THE  CASE  WHEN  THE  REFRACTING  EDGES  OF  THE  PRISMS  ARE  ALL 

PARALLEL,  AND  THE  RAY  LIES  IN  A  PRINCIPAL  SECTION 

COMMON  TO  ALL  THE  PRISMS. 

91.  A  series  of  transparent  optical  media  separated  from  each  other 
by  plane  refracting  surfaces  constitutes  a  system  of  prisms;  the  second 
face  of  one  prism  being  at  the  same  time  the  first  face  of  the  following 
prism  of  the  series.  If  there  are  m  +  i  plane  refracting  surfaces,  we 
shall  have  a  system  of  m  prisms.  We  shall  assume  here  (as  is  almost 
invariably  the  case  in  actual  practice)  that  the  edges  of  the  prisms  are 
parallel  straight  lines;  accordingly,  any  plane  perpendicular  to  this 
system  of  parallel  lines  will  be  a  principal  section  common  to  all  the 
prisms. 

In  the  diagram  (Fig.  45)  the  plane  of  the  paper  is  supposed  to  be 
a  plane  of  a  principal  section  of  the  prism-system.  The  straight  line 
LlBl  represents  the  path  of  a  ray  incident  at  the  point  Bl  on  the  first 
refracting  plane  /zt  of  the  series  of  m  refracting  planes.  The  problem 
is  to  determine  the  path  of  the  emergent  ray  BmBm+l  after  the  ray  has 
been  refracted  in  succession  at  each  of  the  m  refracting  planes. 

The  absolute  indices  of  refraction  of  the  successive  media  traversed 
by  the  ray  will  be  denoted  by  nlt  n'lt  n'2,  etc.,  so  that  nk^  will  denote 
the  index  of  refraction  of  the  kth  medium,  and,  consequently,  nm  will 
denote  the  index  of  refraction  of  the  medium  into  which  the  ray  emerges 
after  the  mth  refraction.  In  actual  prism-systems  it  is  usually  the 

1L.  BURMESTER:  Homocentrische  Brechung  des  Lichtes  durch  das  Prisma:  Zft.  /. 
Math.  u.  Phys.,  xl.  (1895),  65-90. 


112 


Geometrical  Optics,  Chapter  IV. 


92. 


case  that  every  other  medium  of  the  series  is  air,  and  almost  invariably 
the  first  and  last  media  are  air,  so  that  nv  =  n'2  =  n'^  =  •  •••  =  rim\  but 


FIG.  45. 

PATH  OF  A  RAY  IN  A  COMMON  PRINCIPAL  SECTION  OF  A  SYSTEM  OF  PRISMS  WITH  THEIR 
REFRACTING  EDGES  ALL  PARALLEL. 

for  the  sake  of  generality  we  shall  not  assume  here  (except  in  special 
cases)  that  any  two  of  the  series  of  media  are  the  same. 

The  points  in  the  diagram  where  the  refracting  edges  of  the  prisms 
meet  the  common  principal  section  are  designated  as  Vlt  V2,  etc.; 
thus,  the  point  where  the  refracting  edge  of  the  &th  prism  (that  is, 
the  straight  line  in  which  the  two  refracting  planes  M*  and  M*+I  inter- 
sect) meets  the  principal  section  will  be  designated  as  the  point  Vk. 

92.  Construction  of  the  Path  of  the  Ray.  The  path  of  a  ray 
through  a  system  of  prisms  can  be  constructed  geometrically  by  re- 
peated applications  of  the  construction  of  the  path  of  a  ray  through 
a  single  prism  (§66).  Thus,  with  centre  at  the  point  F!  and  with 
radii  rlf  n^rjn^  and  n'2rljn[  (where  rl  denotes  any  arbitrary  length) 
describe  the  arcs  of  three  concentric  circles;  and  through  Vl  draw  a 
straight  line  parallel  to  the  given  incident  ray  L^  meeting  the  circle 
of  radius  n^r^n^  in  the  point  Gr  Through  Gl  draw  a  straight  line 
perpendicular  to  the  first  refracting  plane  /^  and  meeting  the  circle 
of  radius  rl  in  the  point  Hj ;  and  from  Hl  draw  a  straight  line  perpen- 
dicular to  the  second  refracting  plane  ju2  and  meeting  the  circle  of 
radius  n'2rl/n'l  in  the  point  J^  and  draw  the  straight  lines  VlHl  and 
.  Similarly,  with  F3  as  centre  and  with  radii  r2,  n'2r2/n'z  and 
n'3  describe  the  arcs  of  three  concentric  circles,  and  through  F3 


§  93.]  Refraction  Through  a  Prism  or  Prism-System.  113 

draw  a  straight  line  parallel  to  VlJl  meeting  the  circle  of  radius  n'2r2/n3 
in  the  point  G3.  The  points  H3  and  73  are  determined  by  drawing  the 
straight  lines  G3H3  and  H3J3  perpendicular  to  the  refracting  planes 
/*3  and  /J4,  respectively.  Finally,  draw  the  straight  lines  V3HS  and 
F3/3.  Having  performed  this  construction  as  often  as  necessary,  we 
can  construct  the  path  of  the  ray  through  each  prism  in  succession. 
Thus,  we  must  draw  BLB2  parallel  to  V1HIJ  B2B3  parallel  to  VlJl  (or 
toF3G3),  BSB4  parallel  to  F3tf3,  B4B5  parallel  to  F3/3,  etc.;  where 
Bk  designates  the  point  of  incidence  of  the  ray  at  the  kth  refracting 
plane  vk. 

93.  Formulae  for  the  Trigonometrical  Calculation  of  the  Path  of 
the  Ray  through  the  System  of  Prisms.  The  refracting  angle  of  the 
kth  prism  of  a  system  of  prisms  is  the  angle  through  which  the  re- 
fracting plane  /z&  has  to  be  turned  about  the  refracting  edge  of  the 
prism  in  order  that  this  plane  shall  be  brought  to  coincide  with  the 
plane  nk+l.  This  angle  will  be  denoted  by  /3A;  thus, 


The  angles  of  incidence  and  refraction  at  the  kth  refracting  plane  //ft 
will  be  denoted  by  ak,  a'k.  Thus,  if  the  straight  line  O1t_lBkOk  is  the 
normal  to  the  plane  //A  at  the  incidence-point  Bk,  then 

Z  Ok_lBkBk_l  =  ak,      Z  OkBkBk+l  =  a'k. 

The  angle  of  deviation  at  the  kth  refracting  plane  pk  is  the  acute  angle 
through  which  the  straight  line  BkBk+l  must  be  turned  about  the 
point  Bk  in  order  that  BkBk+l  may  have  the  same  direction  as  Bj^^B^ 
This  angle  will  be  denoted  by  the  symbol  ek.  The  total  deviation, 
denoted  by  the  symbol  e  (without  any  subscript)  ,  is  the  angle  through 
which  the  emergent  ray  must  be  turned  in  order  that  its  direction 
may  be  the  same  as  that  of  the  incident  ray.  Thus,  in  case  there  are 
m  refracting  surfaces, 

k=m 
€  =  ZX 

All  these  angular  magnitudes  are  reckoned  positive  or  negative  ac- 
cording as  the  rotation  is  counter-clockwise  or  clockwise. 

Accordingly,  for  calculating  the  path  of  a  ray  through  a  system  of 
prisms,  consisting  of  m  refracting  planes,  the  refracting  edges  of  the 
prisms  being  all  parallel,  and  the  ray  lying  in  a  principal  section  com- 


114  Geometrical  Optics,  Chapter  IV.  [  §  94. 

mon  to  all  the  prisms,  we  have  the  following  system  of  equations:1 

I  II  in 

w[-sin  a{  — ^'Sin  alf  '_/Q  e\  —  <x\~0in     GO 

n~  •  sin  a'9  =  n(  -  sin  a,,  /  e2  =  a2  ~~  «2 >     (^2) 


WA  •  sin  aA  =  nk_l  •  sin  a^       *       *-1  €^  =  0:^  —  0;^., 


k=m 

Total  Deviation  =  e  =  ]C  < 


(43) 


Here  the  term  Z^fc™"1  ft  =  angle  between  the  first  and  last  (or  wth) 
refracting  planes;  and  if  we  denote  this  angle  by  ft,  we  can  write: 


94.     Condition  that  the  Total  Deviation  shall  be  a  Minimum.     The 

total  deviation  e  of  a  ray  refracted  through  a  given  system  of  prisms 
will  be  a  minimum  when  the  ray  is  incident  on  the  first  refracting 
plane  at  an  angle  ax  determined  by  the  condition  de/d^  =  o;  it  being 
assumed  that  the  conditions  dzejda\  >  o  and  e  >  o  are  also  fulfilled. 
According  to  the  equation  above,  the  condition  de/d^  —  o  is  equiva- 
lent to: 


and  in  order  to  express  da'm  as  a  function  of  da^  we  employ  the  equa- 
tions in  columns  I  and  II  of  the  system  of  equations  (43).  Thus, 
differentiating  each  of  these  equations,  we  obtain: 

,          ^ 

oa,  =  —7 


cos  at 

,       n{  cos  a2 

da*  =  —  7  •  --  > 
2       n2  cos  a2 


!See  A.  GLEICHEN:  Ueber  die  Brechungdes  Lichtes  durch  Prismen:  Zft.  f.  Math.  u. 
Phys.,  xxxiv.  (1889),  161-176.  Also,  S.  CZAPSKI:  Theorie  der  optischen  Instrumente  nach 
ABBE  (Breslau,  1893),  S.  137.  H.  KAYSER:  Handbuch  der  Spectroscopie,  Bd.  I  (Leipzig, 
1900),  S.  272.  F.  LOEWE:  Die  Prismen  und  die  Prismensysteme:  Chapter  VIII  of  Die 
Theorie  der  optischen  Instrumente,  Bd.  I,  herausgegeben  von  M.  VON  ROHR  (Berlin,  1904), 
S.  421. 


§  95.]  Refraction  Through  a  Prism  or  Prism-System.  115 

Combining  these  equations,  we  obtain  for  a  system  of  m  refracting 
planes : 

,       nl  cos  al  •  cos  «2 '  '  *  cos  am 

dam  —  — 7 ~, —     — f — T  da,. 

nm  cos  at  •  cos  a2  -  •  -  cos  am 

Hence,  putting  dal  =  da'm,  we  obtain  as  the   Condition  of  Minimum 
Deviation: 

cos  «!  •  cos  a2 '  -  •  cos  am      nm 

cos  a[  -  cos  «2  *  *  *  cos  am      nl 
This  formula  may  be  written  in  abbreviated  form  as  follows: 

k=m  k=m 

HI  II  cos  ak  =  n'm  II  cos  ak,  (44) 


where  the  symbol  II  is  used  to  denote  the  product  of  a  series  of  terms. 
In  the  special  case  when  the  first  and  last  media  are  the  same  (n^  =  n'm), 
the  condition  of  Minimum  Deviation  is: 

k=m  k=m 

II  cos  ak  =  II 


ART.  29.     REFRACTION,  THROUGH  A  SYSTEM  OF  PRISMS,  OF  AN  INFINITELY 

NARROW,  HOMOCENTRIC  BUNDLE  OF  INCIDENT  RAYS:  THE  CHIEF 

RAY  THEREOF  LYING  IN  A  PRINCIPAL  SECTION 

COMMON  TO  ALL  THE  PRISMS. 

95.  An  infinitely  narrow,  homocentric  bundle  of  incident  rays  is 
refracted  through  a  system  of  prisms  with  their  refracting  edges  all 
parallel,  the  chief  ray  of  the  bundle  lying  in  a  Principal  Section  com- 
mon to  all  the  prisms;  so  that  the  meridian  sections  of  the  bundles 
of  incident  and  refracted  rays  coincide  with  the  plane  of  the  principal 
section,  whereas  the  planes  of  the  sagittal  sections  intersect  in  straight 
lines  parallel  to  the  prism-edges.  The  system  of  prisms  being  given, 
the  problem  is  to  determine  the  positions  on  the  chief  emergent  ray 
u'm  of  the  I.  and  II.  Image-Points  S'm  and  3^  corresponding  to  a  given 
Object- Point  Sl  lying  on  a  given  incident  chief  ray  u±\  the  number 
of  refracting  planes  being  denoted  by  m. 

Geometrical  Construction  of  the  I.  and  II.  Image-Points.  In  the  dia- 
gram (Fig.  46)  only  the  first  three  prisms  of  the  system  are  represented. 
The  chief  ray  uv  of  the  bundle  of  incident  rays  meets  the  first  refract- 
ing plane  at  the  point  Bl ;  and  the  path  of  this  ray  through  the  system 
of  prisms  must  be  constructed  as  explained  in  §  92.  If  on  the  straight 
line  FX  drawn  parallel  to  the  straight  line  B1B2  (or  u()  we  take  a  point 


116 


Geometrical  Optics,  Chapter  IV. 


[§95. 


Z;,  and  construct,  as  in  §  76,  the  corresponding  points  Zx  and  Z;  lying 
on  FA  and  V}z'2  drawn  parallel  to  u^  and  B2B^  (or  u'2),  respectively; 
and  if,  in  the  same  way,  to  a  point  K'3  taken  on  the  straight  line  F3£3 


drawn  parallel  to  the  straight  line  BSB±  (or  u'3),  we  construct  the  cor- 
responding points  K'2  and  K'±  lying  on  the  straight  lines  Vsk2  and  V3k'4 
drawn  parallel  to  B2BS  (or  u'2)  and  -B^,  respectively;  we  can  con- 
struct the  I.  Image-Point  £4  on  the  chief  ray  u\  of  the  bundle  of  rays 
refracted  at  the  fourth  refracting  plane  which  corresponds  to  any  as- 


§  96.]  Refraction  Through  a  Prism  or  Prism-System.  117 

sumed  Object-  Point  S1  lying  on  the  chief  incident  ray  ult  as  follows: 
Through  Sl  draw  a  straight  line  parallel  to  ZlZ'l  meeting  u{  in  the 
point  S(  ;  through  S(  draw  a  straight  line  parallel  to  Z{  Z2  meeting  u'2 
in  the  point  5^;  through  S'2  draw  a  straight  line  parallel  to  K2K'B 
meeting  u's  in  the  point  S'3;  and,  finally,  through  S'3  draw-  a  straight 
line  parallel  to  K'3K'4  meeting  the  chief  ray  u'4  in  the  I.  Image-Point 
S'4  corresponding  to  the  Object-Point  Sl  on  the  chief  incident  ray  «,. 

In  order  to  construct  the  II.  Image-Point  S'^  we  draw  SJ5{  perpen- 
dicular to  the  first  refracting  plane  MI  and  meeting  u(  in  the  point  S{  ; 
and  draw  S[S2  perpendicular  to  the  second  refracting  plane  n2  and 
meeting  u'2  in  the  point  S'2,  and  draw  3J3J  perpendicular  to  the  third 
refracting  plane  /J3  and  meeting  u'3  in  the  point  S'3',  and,  finally,  we 
draw  S'sS't  perpendicular  to  /*4  and  meeting  the  emergent  ray  u'4  in 
the  II.  Image-Point  3^  corresponding  to  the  Object-  Point  Sl  on  the 
chief  incident  ray  uv 

If  we  have  more  than  four  refracting  planes,  we  have  merely  to 
continue  the  construction  as  above-indicated  until  we  have  constructed 
the  I.  and  II.  Image-Points  S'm  and  ~S'm  lying  on  the  chief  emergent 
ray  corresponding  to  the  homocentric  Object-Point  Sl  on  the  chief 
incident  ray  ur 

Applying  here  also  the  results  which  were  found  in  §  63  ,  we  can  say  : 

Corresponding  to  a  Range  of  Homocentric  Object-  Points  Plf  Qlt  Rlt-  •  - 
on  an  incident  chief  ray  u^  which  is  refracted,  in  a  principal  section, 
through  a  System  of  Prisms,  we  have  on  the  chief  emergent  ray  um  a 
Similar  Range  of  I.  Image-  Points  P'm1  Q'm,  R'm,-  -  •  and  a  Similar  Range 
of  II.  Image-  Points  P'm,  ~Q'm,  R'm,--. 

96.  Formulae  for  Calculation  of  the  Positions  on  the  Chief  Emer- 
gent Ray  of  the  I.  and  II.  Image-Points.  The  linear  magnitudes  in 
the  following  equations  will  be  denoted,  in  accordance  with  our  previous 
notation,  by  the  following  system  of  symbols: 

The  distances  from  the  incidence-point  Bk  (Fig.  47),  measured  along 
the  chief  ray,  before  and  after  refraction  at  the  &th  surface,  of  the 
I.  and  II.  Image-Points  will  be  denoted  as  follows: 


The  length  of  the  ray-path  within  the  kth  prism,  or  the  distance 
measured  along  the  chief  ray  %  between  the  &th  and  the  (&+i)th  re- 
fracting planes,  will  be  denoted  as  follows: 


The  definitions  and  symbols  of  the  angular  magnitudes  are  the  same 
as  those  given  in  §  93. 


118 


Geometrical  Optics,  Chapter  IV. 


[§96. 


We  shall  assume  that  the  system  of  prisms  is  formed  by  m  plane 
refracting  surfaces  and  that  the  edges  of  the  (m  —  i)  prisms  are  all 
parallel. 

According  to  formulae  (19)  and  (20)  of  §  59,  we  have  for 
The  Meridian  Rays  After  Refraction  at  the  kth  Plane: 


n     cos  a 


'&-! 

V_!  cos  ak 
rik   cosc^ 


(k  =  2,  3,  •  •  •,  w), 


(45) 


In  the  first  and  third  of  these  formulae  we  must  give  k  in  succession 
all  integral  values  from  k  =  i  to  k  =  m;  and  in  the  second  all  integral 


FIG.  47. 
SHOWING  THE  PATH  OF  THE  CHIEF  RAY  THROUGH  THE  £TH  PRISM  OF  A  SYSTEM  OF  PRISMS. 

values  from  k  =  2  to  k  =  m.     It  may  be  observed  also  that  n^  =  nr 
Eliminating  sk  from  the  first  two  of  equations  (45),  and  abbreviating 
by  writing: 

_  nk     cos2  ak 
k  ~  nk_lcos2  ak 


§  96.]  Refraction  Through  a  Prism  or  Prism-System, 

we  obtain: 

and  by  successive  applications  of  this  formula: 


119 


Rk  •  sk_2  —  (R^  -  Rk  -  8k_2  +  Rk  - 


=  Rk_ 


k_2 


R 


Rk_1  -  Rk 


=  etc.,  etc. 
Thus, 

sk  =  (Rl-R2 
-(R2-R3- 

and,  hence: 


(It  should  be  remarked  that  s'Q  —  sl  and  50  =  o.) 
Now 


k=m  k=m—L    (  r=m  1 

;-vii«.-  E  v  n^  • 

A=l  A=l       I          r=k+l          ] 


and 


=i  cos 


cos 


accordingly,  we  obtain  finally  the  following  formula  for  calculating 
the  position  of  the  I.  Image-Point  S'm  on  the  chief  emergent  ray  um: 


cos2 


cos'  ": 


Similarly,  according  to  formulae  (21)  and  (22)  of  §  60,  we  have  for 
The  Sagittal  Rays  After  Refraction  at  the  kth  Surface: 


(k  =  i,  2,  •  ••,  m), 
(k  =  2,  3,  •  •  •,  m), 
(k  =  1,2,  -',m). 


(47) 


Eliminating  5^.  from  the  first  two  of  these  equations,  we  obtain: 


120  Geometrical  Optics,  Chapter  IV.  [  §  97. 

and  by  successive  applications  of  this  formula  : 


_.      nk_2 
etc.,  etc., 


and,  hence,  the  formula  for  calculating  the  position  of  the  II.  Image 
Point  15'm  on  the  chief  emergent  ray  u'm  is  as  follows: 

'  k=m-l  g 

97.    The  Convergence-Ratios  of  the  Meridian  and  Sagittal  Rays. 
The  Convergence-Ratio  of  the  Meridian  Rays, 


«      da,  J 
is  easily  obtained  from  the  third  of  formulae  (45)  ;  for,  since 


we  find  immediately: 

-  (49) 


In  the  same  way  the  Convergence-Ratio  of  the  Sagittal  Rays, 

7   _^ 
Z"  "  rfX,  ' 

is  found  immediately  by  the  third  of  formulae  (47)  ;  for,  since 

^X^_!  =  d\k1 
we  obtain: 


In  viewing  through  a  system  of  prisms  the  II.  Image  of  an  illumi- 
nated slit,  with  its  length  adjusted  parallel  to  the  prism-edges,  we  see 
by  formula  (50)  that,  provided  nm  =  nlt  the  apparent  length  of  the 
slit  will  not  be  altered;  whereas,  in  general,  the  apparent  breadth  of 
the  slit-image  will  be  different  from  that  of  the  slit  itself,  depending 


§  98.]  Refraction  Through  a  Prism  or  Prism-System.  121 

on  the  direction  of  the  chief  incident  rays,  according  to  formula  (49)  ; 
exactly  as  was  found  to  be  the  case  in  viewing  the  slit-image  through 
a  single  prism  (§§79  and  86). 

98.  Formula  for  the  Astigmatic  Difference.  The  expression  for 
the  astigmatic  difference  of  the  bundle  of  emergent  rays  correspond- 
ing to  an  infinitely  narrow,  homocentric  bundle  of  incident  rays  re- 
fracted, in  a  principal  section,  through  a  system  of  prisms  consisting 
of  m  refracting  planes  may  be  found  from  formulae  (46)  and  (48),  as 
follows  : 


cos  ak 

k=m—l 


The  magnitude  of  the  astigmatic  difference  depends  therefore  not  only 
on  the  constants  of  the  prism-system  and  on  the  direction  of  the  chief 
incident  ray  and  on  the  position  of  the  Object-Point  on  this  ray,  but 
also  on  the  lengths  of  the  ray-paths  within  the  prisms;  in  general,  it 
will  be  different  from  zero. 

The  condition  that  the  astigmatic  difference  shall  be  independent 
of  the  ray-distance  (sj  of  the  Homocentric  Object-Point  Sl  from  the 
incidence-point  Bl  of  the  chief  ray  at  the  first  refracting  plane  is 
evidently : 

k=m  Ie=m 

H  cos  ak  =  JJ  cos  dk\ 


=\ 


which,  provided  the  medium  of  the  emergent  rays  is  the  same  as  that 
of  the  incident  rays,  is  the  condition  that  the  chief  ray  shall  traverse 
the  prism-system  with  minimum-deviation  (§  94). 

The  magnitude  of  the  astigmatic  difference  depends  essentially,  as 
was  remarked  above,  on  the  lengths  dk  of  the  ray-paths  within  the 
prisms.  If  the  condition  expressed  by  the  last  equation  is  fulfilled, 
and  if,  in  addition,  each  of  the  (m  —  i)  values  of  5k  is  infinitesimally 
small  (a  condition  which  is  hardly  practicable  geometrically,  especially 
if  there  is  more  than  one  prism),  the  astigmatic  difference  vanishes. 

,For  finite  values  of  5A,  the  magnitude  of  the  astigmatic  difference 
is  of  comparatively  less  and  less  importance,  the  farther  the  Homo- 
centric  Object-Point  5L  is  removed  from  the  incidence-point  Br  Thus, 
when  the  incident  rays  are  a  bundle  of  parallel  rays  (sl  =  oo),  the 
ratio  (sm  —  s'm)/sl  is  vanishingly  small;  so  that,  practically,  the  astig- 
matism is  equal  to  zero  in  this  case.  These  results  are  exactly  anal- 


122  Geometrical  Optics,  Chapter  IV.  [  §  99. 

ogous  to  those  which  were  obtained  in  the  case  of  a  single  prism  (§  §  80 
and  foil.). 

99.  Homocentric  Refraction  through  a  System  of  Prisms.  If  the 
astigmatic  difference  of  the  bundle  of_emergent  rays  is  equal  to  zero, 
the  I.  and  II.  Image-Points  S'm  and  5'm  will  coincide  in  a  single  point 
S'n  on  the  chief  emergent  ray  um;  and  in  this  case  the  image  of  a  point- 
source  Si  on  the  chief  incident  ray  u^  will  be  a  point  l!m.  Thus,  ex- 
actly as  in  Art.  25,  where  the  special  case  of  homocentric  refraction 
through  a  single  prism  was  investigated,  the  condition  that  the  astig- 
matic difference  shall  vanish  is  found  by  putting  sm  -  s'm  =  o  in  for- 
mula (51);  whereby  we  obtain: 


a;\ 

J  ' 


where  Sx  designates  the  Homocentric  Object-Point  on  the  chief  inci- 
dent ray  u^  to  which  corresponds  the  Homocentric  Image-Point  l!m 
on  the  chief  emergent  ray  u'm.  This  formula  gives  the  distance  BlZl 
as  a  unique  function  of  the  angle  of  incidence  a^  of  the  chief  incident 
ray  u^  and  the  ray-lengths  5k  from  one  surface  to  the  next;  so  that 
precisely  as  in  the  case  of  a  single  prism  (§  83),  we  have  also  here  the 
following  statement: 

On  every  incident  chief  ray  ult  refracted  in  a  principal  section  through 
a  system  of  prisms  with  their  edges  all  parallel,  there  is,  in  general,  one, 
and  only  one,  Object-  Point  St  to  which  on  the  chief  emergent  ray  um  there 
corresponds  a  Homocentric  Image-  Point  2^. 

It  is  easy  to  show  likewise  by  the  same  methods  as  were  used  in 
the  case  of  a  single  prism  (§§83,  foil.)  that  Object-  Points,  lying  on 
parallel  incident  chief  rays,  refracted,  in  a  principal  section,  through  a 
system  of  prisms  with  their  edges  all  parallel,  to  which  on  the  parallel 
emergent  chief  rays  correspond  Homocentric  Image-  Points,  are  ranged 
along  a  certain  straight  line  a^  and  the  Homocentric  Image-  Points  are 
ranged  also  along  a  straight  line  am,  which  may  be  regarded  as  the  emer- 
gent ray  corresponding  to  the  incident  ray  at. 

The  construction  of  the  Homocentric  Image-Point  2^  on  the  chief 
emergent  ray  um  and  of  the  corresponding  Homocentric  Object-Point 
Sx  on  the  chief  incident  ray  u\  is  performed  by  a  method  entirely 
analogous  in  every  detail  to  the  method  given  for  the  case  of  a  single 
prism  (§  84).  The  system  of  Object-Points  PUji,  Pv,i,  etc.,  lying  on 
parallel  incident  chief  rays  «„  v±,  etc.,  may  be  denoted  as  the  system 


§100.] 


Refraction  Through  a  Prism  or  Prism-System. 


123 


i}^  and  this  system  of  Object- Points  is  in  affinity  with  the  system  of 
corresponding  I.  Image-Points  (system  r/'J  and  with  the  system  of  cor- 
responding II.  Image-Points  (system  rj'm);  and  the  straight  line  a'm, 
whereon  lie  the  double-points  of  the  systems  T\m,  rj'mt  or  the  Homocentric 
I  mage1- Points  of  this  system  of  parallel  chief  rays,  is  the  affinity-axis 
of  the  systems  i]'m  and  i{m.  This  straight  line  a'm  can  be  constructed, 
therefore,  by  finding  the  points  of  intersection  of  any  two  pairs  of  cor- 
responding straight  lines  of  the  systems  i{m  and  ~rj'm.1 


--C 


ART.   30.     PATH   OF   A   RAY   REFRACTED    OBLIQUELY   THROUGH   A   PRISM. 

100.  Construction  of  the  Path  of  the  Ray.  In  the  figure  (Fig.  48) 
the  prism  is  shown  in  oblique  parallel  projection  with  its  refracting 
edge  Vy  vertical  and  lying  in  the  plane  of  the  paper  which  is  supposed 
to  coincide  with  the  plane  of  the  second  face  of  the  prism.  The  con- 
struction of  the  ray  refracted  at  the  first  face,  corresponding  to  a  ray 
LlBl  incident  obliquely 
on  this  face  at  the  point 
Bv  is  exactly  similar  to 
the  construction  given 
in  §  34.  The  plane  con- 
taining the  incidence- 
point  Bl  and  perpen- 
dicular to  the  prism-edge 
will  be  the  plane  of  a 
principal  section.  The 
straight  lines  B^M  and 
B±N  normal  to  the  two 
faces  /Zj  and  ju2,  respec- 
tively, will  lie  in  this 
principal  section.  Let 
the  incident  ray  LlBl 
prolonged  meet  the  sec- 
ond face  of  the  prism 
in  the  point  designated 
by.D;  so  that  BJ) M  is  the  plane  of  incidence  at  the  first  face.  If 
the  triangle  BfiM  is  revolved  about  MD  as  axis  until  it  comes 
into  the  plane  of  the  paper,  the  point  Bl  will  arrive  at  a  point  C 
lying  on  the  straight  line  drawn  from  N  perpendicular  to  MD  and 
at  a  distance  from  M  which  will  be  the  real  distance  of  Bl  from  M. 

1  See  L.  BURMESTER:  Homocentrische  Brechung  des  Lichtes  durch  das  Prisma:  Zft. 
f.  Math.  u.  Phys.,  xl.  (1895),  65-90. 


FIG.  48. 

CONSTRUCTION  OF  THE  PATH  OF  A  RAY  REFRACTED 
OBLIQUELY  THROUGH  A  PRISM.  Vy  represents  the  refract- 
ing edge  of  the  prism,  and  the  plane  of  the  paper  is  supposed 
to  coincide  with  the  second  face  of  the  prism. 


124  Geometrical  Optics,  Chapter  IV.  [  §  101. 

In  the  diagram,  as  drawn,  the  real  distance  of  B^N  is  supposed  to 
be  twice  the  length  of  Bfl  in  the  diagram.  With  the  point  C  as 
centre  and  with  radius  equal  to  n(-  CD/n^  describe,  in  the  plane  of 
the  paper,  the  arc  of  a  circle  meeting  the  straight  line  drawn  through 
D  parallel  to  the  straight  line  CM  in  the  point  R ;  and  draw  the  straight 
line  CR  meeting  MD  in  the  point  J52;  then  Z.MCD  =  alt  /.MCB2 
=  «j;  and  the  straight  line  B^B2  will  represent  in  the  diagram  the 
path  of  the  ray  within  the  prism. 

The  plane  of  incidence  at  the  second  face  of  the  prism  is  the  plane 
B^B2N,  and  if  we  revolve  the  triangle  B1NB2  about  NB2  as  axis 
until  it  comes  into  the  plane  of  the  paper,  the  point  Bl  will  arrive 
at  a  point  0  on  the  straight  line  NO  perpendicular  to  NB2,  the  length 
of  ON  being  the  real  length  of  B^,  and  Z  B2ON  =  «2.  With  the 
point  0  as  centre  and  with  radius  OT  —  n'2-OB2/n(,  describe  in  the 
plane  of  the  paper  the  arc  of  a  circle  meeting  the  straight  line  drawn 
from  B2  parallel  to  the  straight  line  NO  in  the  point  designated  by  T\ 
then  Z  TON  =  a'2.  If  G  designates  the  point  of  intersection  of  the 
straight  lines  OT  and  NB2,  then  BVG  will  be  the  direction  of  the 
emergent  ray,  and  the  straight  line  B2Q2  drawn  parallel  to  B^G  will 
represent  in  the  figure  the  path  of  the  emergent  ray. 

101.  Formulae  for  Calculating  the  Path  of  a  Ray  Refracted  through 
a  Prism  Obliquely.  When  the  path  of  the  ray  does  not  lie  in  a 
principal  section  of  the  prism,  we  must  employ  the  formulae  of  §§  32,  33. 
Thus,  if  the  inclinations  to  the  plane  of  a  principal  section  of  the 
incident  ray  LlBl  and  of  the  ray  B}B2  refracted  at  the  first  face  of  the 
prism  are  denoted  by  the  symbols  77  L  and  rj{,  respectively;  and,  similarly, 
if  the  inclination  of  the  emergent  ray  B2Q2  is  denoted  by  rj2,  we  have: 

Wj-sin  rjl  =  n[-sm  rj J, 

n{  -  sin  77  j  =  n'2  •  sin  ri2 ; 
and,  hence, 

Wj-sin  rjl  =  n2-sin  rj2.  (53) 

If,  as  is  usually  the  case,  nv  =  n'2,  we  shall  have  ^  =  »?2;  accordingly, 
when  the  prism  is  surrounded  on  both  sides  by  the  same  medium,  the 
incident  and  emergent  rays  are  equally  inclined  to  the  plane  of  the  prin- 
cipal section. 

Moreover,  if,  as  in  §33,  the  symbols  ylt  y[  and  72,  y'2  denote  the 
angles  which  the  normals  to  the  two  faces  of  the  prism  make  with  the 
projections  in  a  principal  section  of  the  incident  and  refracted  rays 


§  102.]  Refraction  Through  a  Prism  or  Prism-System.  125 

at  the  first  and  second  faces,  respectively,  we  have  also: 

nl  •  cos  77!  •  sin  yl  =  n(  •  cos  77 [  -  sin  y( , 

n[-cos  77i-sin  y2  =  n'2-  cos  773 -sin  7^  ^  (54) 

T2 

These  formulae  (53)  and  (54)  enable  us  to  determine  completely  the 
emergent  ray  corresponding  to  a  given  incident  ray. 

If  the  prism  is^  surrounded  by  the  same  medium  on  both  sides,  and 
if  we  write  n  =  n'J^  =  n[/n2t  the  formulae  will  be  simplified  as  follows: 

sin  ifjl  =  n-sin  77 J, 
cos  ^  •smy1  =  n-  cos  77  { -  sin  y( , 

r,-*-*  (5S) 

' •  cos  17!  •  sin  72  =  cos  rjl  •  sin  y2. 

102.    Deviation  (D)  of  Ray  Obliquely  Refracted  through  a  Prism. 

If  £  denotes  the  Deviation  of  the  so-called  "Projected  Ray",  that  is, 
the  angle  between  the  projections,  on  the  plane  of  a  principal  section, 
of  the  incident  and  emergent  rays,  then  evidently: 

E  =  yl-y2-p;  (56) 

an  equation  which  is  precisely  similar  to  the  last  of  equations  (25). 
We  proceed  now  to  determine  the  total  deviation  (D)  of  a  ray  obliquely 
refracted  through  a  prism  in  terms  of  this  angle  E.1 

Around  a  point  V  (Fig.  49)  of  the  prism-edge  yVy  as  centre,  describe 
a  spherical  surface,  and  draw  the  radii  Vzl  and  Vz'2  parallel  to  the 
incident  and  emergent  rays,  respectively;  and  let  us  choose  here  for 
the  plane  of  the  paper  the  plane  which  is  determined  by  the  straight 
lines  Vz'2  and  yV.  The  plane  perpendicular  to  the  prism-edge  at  V, 
which  is  therefore  the  plane  of  a  principal  section,  is  the  equatorial 
plane,  and  the  great  circles  yz$  and  yz'2y  which  meet  this  plane  per- 
pendicularly at  the  points  A  and  B,  respectively,  are  meridian  circles 
of  the  sphere  described  on  yVy  as  diameter.  Hence,  VA  and  VB 
are  the  projections  in  the  plane  of  the  principal  section  of  the  radii  Vzl 
and  Vz'2,  respectively;  so  that  the  angles  E  and  D  at  the  centre  of 
the  sphere  are  subtended  by  the  arcs  AB  and  z^  respectively,  inter- 
secting each  other  in  a  point  C.  Now  if  we  assume  that  the  prism  is 

1  See  R.  S.  HEATH:  A  Treatise  on  Geometrical  Optics  (Cambridge,  1887),  Art.  29. 


126  Geometrical  Optics,  Chapter  IV. 

surrounded  by  the  same  medium  on  both  sides,  then 


[  §  102. 


whence  it  follows  that  the  arcs  z^A  and  Bz'2  are  equal,  and  the  right- 
angled  spherical  triangles  ACzlt  BCz2 
are  congruent;  and  therefore  arc  AC 
=  arc  CB,  arc  zLC  =  arc  Cz'2.  From  the 
right-angled  triangle  ACz1  we  have: 

cos  (arc  j^C)  =  cos  (arc  A  C)  •  cos  (arc  AzJ , 


or 


D  E  .    . 

cos—  =  cos-  -cos^.          (57) 


FIG.  49. 

DEVIATION  (D)  OF  A  RAY  RE- 
FRACTED OBLIQUELY  THROUGH  A 
PRISM.  Z.  E  is  the  projection  of  Z  D 
on  the  plane  of  a  principal  section 
A  VB ;  y  Vy  being  the  refracting  edge 
of  the  prism. 


This  equation,  together  with  (56) ,  enables 
us  to  compute  the  total  deviation  D  of  a 
ray  obliquely  refracted  through  a  prism. 
I  twill  be  seen  that  the  angle  D  is  always 
greater  than  the  angle  E. 

The  angle  E  will  have  its  minimum 
value  EQ  at  the  same  time  that  the  angle 
D  has  its  minimum  value  DQ ;  and  the 
condition  that  E  shall  be  a  minimum  is 
given  by  the  equations: 


7i  =  -  72 


which  are  derived   exactly  in  the  same 
way  as  the  analogous  equations  (27)  for 

the  case  of  an  actual  ray  traversing  the  prism  symmetrically  in  the 
plane  of  a  principal  section  were  obtained;  only,  we  must  observe 
that  here  for  the  so-called  "projected  ray",  instead  of  n1  we  have  the 
"artificial"  relative  index  of  refraction  (see  §  33): 


cos 


so  that 


sin 


sin- 
2 


§  103.]  Refraction  Through  a  Prism  or  Prism-System.  127 

which  is  analogous  to  the  formula: 


sm- 


which  is  the  third  of  formulae  (27).  Here  e0  denotes  the  minimum  de- 
viation of  a  ray  traversing  a  principal  section  of  the  prism.  Hence,  we 
obtain  : 


sin 

2  _  WT, 

<•„  +  8        n 


sin 


2 

In  case  we  have  «  >  i,  then  7/1  >  i?i  and,  therefore,  n^>  n;  hence, 

.  £0  +  ff,    .   e0  +  /3 
sin  —       -  >  sin  —    —  • 


Therefore,  the  angle  E0  (which  is  the  projection  on  the  plane  of  a 
principal  section  of  the  angle  DQ  of  the  minimum  deviation  of  a  ray  ob- 
liquely refracted  through  the  prism)  is  always  greater  than  the  angle  of 
minimum  deviation  e0  of  a  ray  incident  on  the  prism  at  the  same  angle  o^ 
but  lying  in  a  principal  section;  and,  hence,  DQ  itself  is  greater  than  e0. 

Of  all  the  rays  which  go  through  a  prism,  that  one  which,  lying  in  a  prin- 
cipal section,  traverses  the  prism  symmetrically  will  be  the  least  deviated. 

The  case  when  n  <  i  may  be  discusssed  exactly  in  the  same  way  as 
was  done  when  the  ray  was  in  the  principal  section  (see  §  71). 

103.  The  formulae  given  in  this  article  for  the  path  of  a  ray  ob- 
liquely refracted  through  a  prism  may  properly  be  attributed  to 
BRAVAIS/  although  the  same  results,  in  a  more  general  form,  were 
afterwards  derived  by  geometric  methods  by  REUSCH2  and  CoRNU3 
and,  analytically,  by  STOKES*  and  HooRWEG.5 

1  A.  BRAVAIS:  Notice  sur  les  parhelies  qui  sont  situes  a  la  m£me  hauteur  que  le  soleil: 
Jour,  de   Vec.  polyt.,  xviii.,  cah.  30  (1845),  79.     Memoire  sur    les  halos,  etc.:  Journ.  de 
Vec.  polyt.,  xviii.,   cah.  31    (1847),   27. 

2  E.  REUSCH:  Die  Lehre  von  der  Brechung  und  Farbenzerstreuung  des  Lichts  an  ebenen 
Flaechen  und  in  Prismen  in  mehr  synthetischer  Form   dargestellt:   POGG.  Ann.,  cxvii. 
(1862),    241-284. 

3  A.  CORNU:  De  la  refraction  a  travers  un  prisme  suivant  une  loiquelconque:  Ann.  tc. 
norm.,  (2)  I.  (1872),  255-257. 

4  G.  G.  STOKES:  In  a  "  Note  "  on  a  paper  by  TH.  GRUBB:  Proc.  Roy.  Soc.,  xxii.  (1874), 

309- 

5J.  L.  HOORWEG:  Ueber  den  Gang  der  Lichtstrahlen  durch  ein  Spectroscop:  POGG. 
Ann.,  cliv.  (1875),  423-430. 

See,  also,  A.  ANDERSON:  On  the  maximum  deviation  of  a  ray  of  light  by  a  prism: 
Camb.  Proc.,  ix.  (i896-'8),  195-197. 


128  Geometrical  Optics,  Chapter  IV.  [  §  104. 

The  explanation  of  the  curvature  of  the  lines  of  the  spectrum,  as 
observed  through  a  prism-spectroscope,  which  appears  to  have  been 
remarked  for  the  first  time  in  GEHLER'S  Physikalisches  Woerterbuch, 
is  to  be  found  in  the  fact  that  the  function  denoted  above  by  n^  de- 
pends on  the  inclination  (77,)  of  the  incident  ray  to  the  principal  sec- 
tion of  the  prism.  BRAVAis1  derived  a  formula  for  the  radius  of  cur- 
vature at  the  vertex  of  the  image-line  which  is  given  in  KAYSER'S 
Handbuch  der  Spectroscopie,  Bd.  I  (Leipzig,  1900),  Art.  321;  also,  in 
F.  LOEWE'S  treatise  Die  Prismen  und  die  Prismensysteme.2 

ART.  31.     HOMOCENTRIC  REFRACTION  THROUGH  A  PRISM  OF  AN  INFI- 
NITELY NARROW,  HOMOCENTRIC  BUNDLE  OF  OBLIQUELY 
INCIDENT  RAYS. 

104.  We  propose  now  to  investigate  the  conditions  that  must  be 
satisfied  in  order  that  to  a  narrow,  homocentric  bundle  of  obliquely 
incident  rays  refracted  through  a  prism  there  shall  correspond  a  homo- 
centric  bundle  of  emergent  rays.  The  solution  of  this  problem  was 
given  first  by  BURMESTER,S  whose  geometrical  method  is  the  one  given 
here.  An  analytical  deduction  of  the  same  results,  based  on  HELM- 
HOLTZ'S  formulae  for  the  passage  of  light  through  a  prism  as  given  in 
his  Handbuch  der  physiologischen  Optik,  has  been  given  by  WiLsiNG.4 

When  a  ray  of  light  is  refracted  through  a  prism,  the  plane  of  inci- 

1  A.  BRAVAIS:  Memoiresur  les  halos,  etc.:  Journ.  de  Vic.  polyt.,  xviii.,  cah.  31  (1847), 
1-280. 

2  See  Die  Theorie  der  optischen  Instrumente:  Herausgegeben  von  M.  VON  ROHR,  Bd.  I 
(Berlin,  1904),  p.  429. 

In  the  same  connection,  see  also  the  following: 

L.  DITSCHEINER:  Ueber  die  Kruemmung  von  Spectrallinien  :  Wien.  Ber.,  li.,  II. 
(1865),  368-383.  Notiz  zur  Theorie  der  Spectralapparate:  POGG.  Ann.,  cxxix.  (1866), 
336-340. 

J.  L.  HOORWEG:  as  cited  above. 

H.  v.  JETTMAR:  Zur  Strahlenbrechung  im  Prisma;  Strahlengang  und  Bild  vonleucht- 
enden  zur  Prismenkante  parallelen  Geraden:  35.  Jahresb.  ueber  das  k.  k.  Staatsgymn.  im 
Bez.  Wiens,  1885. 

J.  v.  HEPPERGER:  Ueber  Kruemmungsvermoegen  und  Dispersion  von  Prismen:  Wien. 
Ber.,  xcii.,  II.  (1885),  261-300. 

A.  CROVA:  Etude  des  aberrations  des  prismes  et  deleur  influence  sur  les  observations 
spectroscopiques:  Ann.  chim.  et  phys.,  5,  xxii.  (1881),  513-520. 

W.  H.  M.  CHRISTIE:  Note  on  the  curvature  of  lines  in  the  dispersion  spectrum,  etc.: 
Monthly  Notices  of  the  Roy.  Astr.  Soc.,  xxxiv.  (1874),  263-5. 

W.  SIMMS:  Note  on  a  paper  by  Mr.  CHRISTIE:  Monthly  Not.,  xxxiv.  (1874),  363-'4. 

See  also  KAYSER'S  Handbuch  der  Spectroscopie,  Bd.  I  (Leipzig,  1900),  Arts.  260,  321, 
322  and  323. 

3L.  BURMESTER:  Homocentrische  Brechung  des  Lichtes  durch  das  Prisma:  Zft.f. 
Math.  u.  Phys.,  xl.  (1895),  65-90. 

4  J.  WILSING:  Zur  homocentrischen  Brechung  des  Lichtes  durch  das  Prisma:  Zft.  f. 
Math.  u.  Phys.,  xl.  (1895),  353-361. 


§  104.]  Refraction  Through  a  Prism  or  Prism-System.  129 

dence  at  the  first  face  and  the  plane  of  emergence  at  the  second  face 
will,  in  general,  not  be  coincident;  in  fact,  this  will  be  the  case  only 
when  the  incident  ray  lies  in  the  plane  of  a  principal  section  of  the 
prism,  as  we  have  seen.  To  a  homocentric  bundle  of  incident  rays 
emanating  from  an  Object-Point  Sx  on  the  chief  incident  ray  ut  there 
corresponds  within  the  prism  an  astigmatic  bundle  of  refracted  rays 
whose  chief  ray  is  designated  by  the  symbol  ult  and  which,  therefore, 
we  may  speak  of  as  the  "bundle  u{".  If  the  incidence-point  of  the 
chief  ray  at  the  first  prism-face  is  designated  by  Blt  the  II.  Image-  Plane 
of  the  astigmatic  bundle  u(  will  be  the  plane  of  incidence  u^B^ 
and  the  I.  I  mage-  Plane  will  be  the  plane  which  contains  u(  and  which 
is  perpendicular  to  the  plane  u^B-^u^. 

On  the  other  hand,  let  us  consider  an  astigmatic  bundle  of  rays 
within  the  prism  whose  chief  ray  may  be  designated  as  the  ray  v[,  and 
which,  therefore,  we  shall  call  the  "bundle  v[".  Let  B2  designate  the 
incidence-point  of  this  chief  ray  v{  at  the  second  prism-face.  More- 
over, let  us  assume  that  the  bundle  of  emergent  rays  corresponding  to 
the  bundle  v{  is  a  homocentric  bundle  of  rays  with  its  vertex  at  a  point 
designated  by  Sg.  The  II.  Image-Plane  of  the  astigmatic  bundle  v( 
coincides  with  the  plane  of  incidence  v[  B^'2  of  the  ray  v[  at  the  second 
face  of  the  prism,  and  the  I.  Image-  Plane  of  this  bundle  is  the  plane 
which  contains  the  ray  v(  and  which  is  perpendicular  to  the  plane 


Now,  if  these  two  astigmatic  bundles  u(  and  v(  within  the  prism  are 
identical,  then  the  point  2'2  is  the  homocentric  Image-Point  on  the 
chief  emergent  ray  *4  which  corresponds  to  the  homocentric  Object- 
Point  Sj  on  the  chief  incident  ray  u^  Now  in  order  that  these  two 
astigmatic  bundles  of  rays  shall  be  identical,  it  is  necessary,  in  the  first 
place,  that  the  I.  and  II.  I  mage-  Planes  of  the  two  bundles  shall  be  co- 
incident; which  may  happen  in  either  of  two  ways:  (i)  The  I.  Image- 
Planes  of  the  two  astigmatic  bundles  of  rays  may  be  identical,  and 
also  the  II.  Image-Planes;  in  which  case  the  chief  rays  will  lie  in  the 
plane  of  a  principal  section  of  the  prism;  which  was  the  case  investi- 
gated in  Art.  25  ;  or  (2)  The  I.  Image-Plane  of  one  bundle  of  rays  may 
coincide  with  the  II.  Image-Plane  of  the  other  bundle,  and  this  is  the 
case  that  interests  us  at  present.  In  this  latter  case,  if  also  the  I. 
Image-Point  of  one  bundle  of  rays  coincides  with  the  II.  Image-Point 
of  the  other  bundle,  and  vice  versa,  the  two  astigmatic  bundles  of  rays 
u(  and  v{  will  be  identical  (provided  we  neglect  infinitesimals  of  the 
second  order,  as  is  here  assumed).  Therefore,  in  order  that,  corre- 
sponding to  an  Object-Point  lying  on  a  chief  incident  ray  which  is  ob- 

10 


130  Geometrical  Optics,  Chapter  IV.  [  §  105. 

liquely  refracted  through  the  prism,  we  shall  have  on  the  chief  emer- 
gent ray  a  homocentric  Image-Point,  it  is  necessary,  first  of  all,  that 
the  planes  of  incidence  and  emergence  shall  be  at  right  angles;  that  is, 
if  «lf  u2  designate  the  chief  incident  ray  and  the  corresponding  chief 
emergent  ray,  respectively,  and  if  the  straight  line  B^B2  represents  the 
path  of  the  chief  ray  from  the  first  face  of  the  prism  to  the  second 
face,  the  two  planes  u^B^B^  and  B^B^u2  must  be  perpendicular. 

105.  In  the  accompanying  diagram  (Fig.  50)  the  refracting  edge  of 
the  prism  is  represented  by  the  vertical  straight  line  Vy  lying  in  the 
plane  of  the  paper,  which,  as  in  the  similar  diagram  (Fig.  48),  is  sup- 
posed to  be  the  plane  of  the  second  face  of  the  prism.  From  the  point 
Bv  in  the  first  face  of  the  prism  draw  the  straight  line  B^M  normal  to 
this  face  and  meeting  the  second  face  in  the  point  designated  by  M 
and  the  straight  line  B±  N  normal  to  the  second  face  at  the  point  des- 
ignated by  N;  so  that  B±MN  will  be  the  plane  of  the  principal  sec- 
tion of  the  prism  which  is  passed  through  the  point  Br  On  the 
straight  line  MN  as  diameter,  describe  in  the  plane  of  the  paper  a 
circle,  only  half  of  which  is  shown  in  the  figure;  and  in  the  circum- 
ference of  this  circle  take  any  point  B2,  and  draw  the  straight  lines 
MB2,  NB2,  B±B2.  If  the  straight  line  B^B2  represents  the  path  within 
the  prism  of  the  chief  ray  of  a  bundle  of  rays,  to  which  corresponds  the 
chief  incident  ray  u^B^  and  the  chief  emergent  ray  B2u'2,  then  B^B2M 
will  be  the  plane  of  incidence  at  the  first  face,  and  B}B2N  will  be  the 
plane  of  emergence  at  the  second  face;  and  these  two  planes  will  be 
at  right  angles  to  each  other,  according  to  the  essential  condition 
which  was  found  in  §  104  above. 

In  order  to  construct  the  chief  incident  ray  #x  and  the  chief  emer- 
gent ray  u2  corresponding  to  a  ray  B1B2  (or  «J)  within  the  prism,  we 
proceed  almost  exactly  as  in  §  100.  First,  we  revolve  the  triangle 
B^B2M  around  the  straight  line  MB2  as  axis  until  it  comes  into  the 
plane  of  the  paper;  so  that  the  point  Bl  falls  at  a  point  C  in  the  straight 
line  NB2  whose  real  distance  from  M  will  depend  on  the  scale  of  the 
oblique  parallel  projection.  In  the  figure  as  here  drawn,  the  real 
length  of  BiN  is  twice  its  length  as  actually  shown.  With  the  point 
C  as  centre,  and  with  radius  equal  to  n^  CB2/n{,  describe  in  the  plane 
of  the  paper  the  arc  of  a  circle  meeting  the  straight  line  drawn  from  B2 
parallel  to  the  straight  line  M  C  in  a  point  designated  by  E,  and  let  D 
designate  the  point  of  intersection  of  the  straight  lines  CE  and  MB2; 
then  the  straight  line  BJ)  will  give  the  direction  of  the  chief  incident 
ray  u^  to  which  within  the  prism  corresponds  the  ray  B^B^. 

Again,  revolve  the  triangle  B^NB^  around  the  straight  line  NB2  as 


§  105.] 


Refraction  Through  a  Prism  or  Prism-System. 


131 


axis  until  it  comes  into  the  plane  of  the  paper,  and  let  the  point  des- 
ignated by  O  be  the  impression  in  this  plane  of  the  point  Blf  so  that 
NO  is  the  real  length  of  the  straight  line  BLN,  and  B2O  =B2C\  and 


with  the  point  0  as  centre  and  with  radius  equal  to  n'2^OB2/n'lt  describe 
in  the  plane  of  the  paper  the  arc  of  a  circle  meeting  the  straight  line 


132  Geometrical  Optics,  Chapter  IV.  [  §  106. 

MB2  in  the  point  designated  by  H,  and  draw  the  straight  line  OH 
meeting  NB2  in  a  point  designated  by  G\  then  the  straight  line  B2u2 
drawn  parallel  to  the  straight  line  B^  will  show  the  path  of  the  emer- 
gent ray  u'2  corresponding  to  the  ray  BiB2  within  the  prism. 

106.  Corresponding  to  an  Object-  Point  Sl  lying  on  the  chief  inci- 
dent ray  «1?  the  I.  Image-Point  S{  on  the  chief  ray  u(  of  the  astigmatic 
bundle  of  rays  refracted  at  the  first  face  of  the  prism  can  be  con- 
structed, according  to  formula  (19),  by  means  of  the  following  relation: 


2 

cos 


A  straight  line  drawn  through  Sl  perpendicular  to  the  first  face  of  the 
prism  will  determine  by  its  intersection  with  the  refracted  ray  u(  the 
II.  Image-Point  S[  corresponding  to  5X. 

A  straight  line  drawn  through  S{  perpendicular  to  the  second  face 
of  the  prism  will  determine  by  its  intersection  with  the  chief  emergent 
ray  u'2  the  II.  Image-Point  R2  which  corresponds  to  the  point  S(  (or  R{). 
The  point  S(  is  the  vertex  of  a  pencil  of  rays  lying  in  the  plane  B^B^M 
which  meet  the  second  prism-face  in  points  infinitely  near  to  the  point 
B2  in  the  straight  line  MB2J  and  which,  being  refracted  at  this  face, 
are  transformed  thereby  into  a  pencil  of  rays  with  vertex  at  the  point 
designated  by  R'2. 

If  T2  designates  the  I.  Image-Point  on  the  ray  u2  which  corresponds 
to  the  point  3J  (or  T{)  on  the  ray  ult  this  point  can  be  constructed  by 
the  following  formula: 


cos 


The  point  3(  is  the  vertex  of  a  pencil  of  rays  lying  in  the  plane  BVB2N 
which  meet  the  second  face  of  the  prism  at  points  infinitely  near  to  B2 
in  the  straight  line  NB2J  and  which,  being  refracted  at  this  face,  are 
transformed  into  a  pencil  of  rays  with  vertex  at  the  point  T'2. 

Thus,  to  a  range  of  Object-  Points  S^  •  •  •  lying  on  the  chief  incident 
ray  u\  there  correspond,  therefore  (see  §  63),  on  the  chief  emergent  ray 
u'2  two  similar  ranges  of  Image-Points  T2,  •  •  •  and  R'2,  •  •  •  ;  and  the 
double-point  Z2  of  these  two  similar  ranges  of  points  lying  along  the  ray 
u'2  may  be  constructed  by  a  method  exactly  similar  to  that  given  in  §  84, 
as  follows:  Produce  the  straight  lines  S(R2  and  3(T2  until  they  inter- 
sect in  a  point  F;  the  point  22  will  be  the  point  of  intersection  of  the 
straight  line  YBl  with  the  emergent  chief  ray  u2.  Through  this  point 
22  draw  a  straight  line  parallel  to  B^N  meeting  u(  in  the  point  Sj; 


§  107.]  Refraction  Through  a  Prism  or  Prism-System.  133 

then  the  straight  line  drawn  through  Si  parallel  to  the  straight  line 
SjSj  will  determine  by  its  intersection  with  the  chief  incident  ray  u^ 
the  Object-Point  Sx  to  which  corresponds  the  Homocentric  Image- 
Point  S2. 

The  results  of  this  investigation  may  be  summarized  as  follows: 

On  every  incident  chief  ray  that  meets  the  first  face  of  the  prism  at  a 
point  Bl  and  that  is  refracted  through  the  prism  along  a  generating  line 
of  the  conical  surface  B^c  (where  c  designates  the  circle  described  on 
the  straight  line  MN  as  diameter  —  see  Fig.  50),  there  is  one  single 
Object-  Point  to  which  on  the  emergent  chief  ray  there  corresponds  a 
Homocentric  Image-  Point. 

107.  The  analytical  expression  for  the  position  of  this  unique 
Object-Point  St  on  such  an  incident  chief  ray  u^  may  be  easily  obtained 
as  follows: 

According  to  formulae  (19)  and  (21),  we  have: 


cos2  ot{  —  ,      n{ 

*      *  BtSi  t     B-,SI  =  —  *  BlSl  ', 
cos  <*  « 


cos 


If  we  put 

d1  =  B1B2  =  B^  —  BZS{  =  B^S(  —  B2S{, 
we  obtain: 


-i-t 


whence  we  find  : 


o  -ii  . 

nl  cos2  ^  w, 


Putting  R2T2  =  o,  we  obtain  finally: 

»t        cos2  ttl  (cos2  a2  -  cos2  o;2) 


iii  ~7  o  -  '  --  2  2  2 

WL  COS    OIj  •  COS    Oi2  —  COS    «!  '  COS    (X2 


CHAPTER  V. 

REFLEXION  AND   REFRACTION  OF   PARAXIAL   RAYS  AT  A  SPHERICAL 

SURFACE. 

ART.  32.     INTRODUCTION.     DEFINITIONS,     NOTATIONS,     ETC. 

108.  In  nearly  all  forms  of  optical  apparatus  the  reflecting  and 
refracting  surfaces  are  spherical;  for  a  plane  may  also  be  regarded 
as  a  spherical  surface  of  infinite  radius.  In  our  diagrams  the  centre 
of  the  reflecting  or  refracting  sphere  will  be  designated  by  the  letter 
C  (Fig.  51).  The  straight  line  determined  by  this  point  C  and  another 

point  M  is  called  the  axis  of 
the  spherical  surface  with  re- 
spect to  the  point  M,  and 
the  point  A  where  the  straight 
^x>^  line  M  C  meets  the  refracting 

*^fo.x — *      (or  reflecting)  surface  is  called 

the  vertex  of  the  surface  with 

\  respect  to  the  point  M .     Evi- 

p,G^  51>  dently ,  a  spherical  surface  will 

RAY  INCIDENT  ON  A  SPHERICAL  SURFACE.  be    symmetrical   with  respect 

to  such  an  axis,  and  the  plane 

of  the  diagram  which  contains  the  axis  is  a  meridian  section  of  the 
spherical  surface. 

Consider  now  an  incident  ray  lying  in  this  plane,  and  crossing  the 
axis,  either  really  or  virtually  (see  §  10),  at  the  point  M.  If  the  point 
M  is  situated  in  front  of  the  vertex  A  (that  is,  to  the  left  of  A),  as 
in  the  figure,  the  intersection  of  an  incident  ray  with  the  axis  at  the 
point  M  will  be  a  "real"  intersection;  whereas  if  the  point  M  lies 
beyond  A  (in  the  sense  in  which  the  incident  light  is  propagated,  which 
in  our  diagrams  is  represented  always  as  being  from  left  to  right),  the 
intersection  of  an  incident  ray  with  the  axis  at  the  point  M  will  be 
a  "virtual"  intersection.  If  B  designates  the  position  of  the  point 
where  the  ray  meets  the  spherical  surface,  and  if  on  the  straight  line 
CB  we  take  a  point  N  in  the  medium  of  the  incident  ray,  the  angle  of 
incidence,  defined  as  in  §  14,  will  be  Z.NBM  —  a.  In  the  figure  the 
plane  of  the  paper  represents  the  plane  of  incidence,  and  after  reflexion 
or  refraction  at  the  point  B,  the  path  of  the  ray  will  still  lie  in  this 
plane. 

134 


§  108.]  Reflexion  and  Refraction  of  Paraxial  Rays.  135 

It  will  be  convenient  to  take  the  vertex  A  of  the  spherical  surface 
as  the  origin  of  a  system  of  plane  rectangular  co-ordinates  ;  the  axis 
of  the  spherical  surface,  defined  as  the  straight  line  AC,  being  taken 
as  the  x-axis,  and  the  tangent  to  the  surface  at  its  vertex  A,  in  the 
incidence-  plane,  being  taken  as  the  y-axis.  The  positive  direction  of 
the  x-axis  is  the  same  as  the  direction  which  light  would  pursue  if  this 
line  were  the  path  of  an  incident  ray  (see  §  26).  The  positive  direction 
of  the  ;y-axis  is  found  by  rotating  the  positive  half  of  the  re-axis  about 
the  point  A  through  an  angle  of  90°  in  a  sense  opposite  to  that  of  the 
motion  of  the  hands  of  a  clock;  so  that  in  our  diagrams  where  the 
x-axis  is  represented  as  a  horizontal  line  with  its  positive  direction 
from  left  to  right,  the  positive  direction  of  the  ;y-axis  will  be  vertically 
upwards. 

The  abscissa  of  the  centre  C,  which  we  shall  call  the  radius  of  the 
spherical  surface,  will  be  denoted  by  the  symbol  r\  thus,  AC  *=  r. 
The  radius  r  is  positive  or  negative  according  as  the  centre  C  lies 
beyond  or  in  front  of  the  vertex  A  ;  and  according  as  the  sign  of  r  is 
positive  or  negative,  the  spherical  surface  is  said  to  be  "convex"  or 
"concave". 

From  the  incidence-point  B  draw  BD  perpendicular  to  the  x-axis 
at  the  point  D\  the  ordinate  h  =  DB  is  called  the  incidence-height 
of  the  ray  which  meets  the  spherical  surface  at  the  point  B.  It  will 
be  positive  or  negative  according  as  the  incidence-point  B  is  above 
or  below  the  x-axis. 

The  slope  of  the  ray  is  the  acute  angle  through  which  the  x-axis 
has  to  be  turned  about  the  point  M  in  order  that  it  may  coincide  in 
position  (but  not  necessarily  in  direction)  with  the  rectilinear  path 
of  the  ray.  This  angle  will  be  denoted  by  the  symbol  6;  thus,  in  the 
figure  /.A  MB  =  0.  Here,  as  always  in  the  case  of  angular  magni- 
tudes, counter-clockwise  rotation  is  to  be  reckoned  as  positive.  The 
sign  of  the  angle  6  may  always  be  determined  from  the  following 
relation  : 


The  acute  angle  at  the  centre  C  of  the  spherical  surface  subtended 
by  the  arc  A  B  will  be  denoted  by  the  symbol  <p.  This  angle  is  defined 
as  the  angle  through  which  the  radius  drawn  to  the  incidence-point  B 
must  be  turned  in  order  that  the  straight  line  B  C  may  coincide  with 
A  C;  thus,  <p  =  ABC  A.  According  to  this  definition,  we  shall  have 
always  sin  <p  =  h/r. 


136  Geometrical  Optics,  Chapter  V.  [  §  109. 

From  the  diagram,  we  derive  at  once  the  following  important  re- 
lation : 

a  =  B  +  <p.  (60) 

109.     From  the  diagram,  also,  we  obtain  easily  the  following  re- 
lations: 

DM      DC+CA  +  AM     r(cos<p -i)  +  AM 


BM  = 


cos  6  cos  6  cos  6 


In  the  special  case  when  the  point  of  incidence  B  is  very  near  to  the  vertex 
A  of  the  spherical  surface,  the  angle  of  incidence  a  will  be  correspond- 
ingly small,  as  will  be  also  the  angles  denoted  by  B  and  <p.  Now  if 
these  angles  a,  6  and  <p  are  all  so  small  that  we  may  neglect  the  second 
and  higher  powers  thereof,  and  write  therefore  in  place  of  the  sines 
of  these  angles  the  angles  themselves  and  also  put 

cos  a  =  cos  6  —  cos  <p  —  I , 

obviously,  we  obtain  in  this  case  BM  =  AM.  Under  these  circum- 
stances, the  ray  MB  is  called  a  Paraxial  Ray. 

A  Paraxial  Ray  is  a  ray  which  proceeds  very  near  to  the  axis  of  the 
Spherical  Surface,  which,  therefore,  meets  this  surface  at  a  point  very 
close  to  the  vertex  and  at  nearly  normal  incidence;  the  angles  a,  B  and  <p 
being  all  so  small  that  we  can  neglect  the  second  powers  of  these  angles. 

The  ray  which  proceeds  along  the  #-axis  is  called  the  axial  ray. 

In  this  chapter,  as  well  as  in  several  chapters  following,  we  shall 
be  concerned  with  the  special  case  of  paraxial  rays  only;  that  is,  we 
shall  consider  only  such  rays  as  proceed  within  a  very  narrow  cylin- 
drical region  immediately  surrounding  the  axis  of  the  spherical  sur- 
face which  is  also  the  axis  of  the  cylinder.  The  only  part  of  the  spher- 
ical surface  that  will  be  utilized  for  reflexion  or  refraction  will  be  the 
small  zone  which  has  the  vertex  A  for  its  summit.  We  may  imagine, 
therefore,  that,  physically  speaking,  the  rest  of  the  spherical  surface 
is  abolished  entirely,  or  that  it  is  rendered  opaque  and  non-reflecting 
by  being  painted  over  with  lamp-black;  or  we  may  suppose  that  a 
screen  with  a  small  circular  opening  is  placed  at  right  angles  to  the 
axis  with  the  centre  of  the  opening  on  the  axis  just  in  front  of  the  ver- 
tex A  of  the  spherical  surface,  so  that  only  such  rays  as  proceed  through 
this  opening  and  are  incident  on  the  spherical  surface  at  points  very 
close  to  the  point  A  will  undergo  reflexion  or  refraction  at  the  spheri- 
cal surface. 


§  no.] 


Reflexion  and  Refraction  of  Paraxial  Rays. 


137 


I.   REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  MIRROR. 

ART.  33.     CONJUGATE  AXIAL  POINTS  IN  THE  CASE  OF  RELEXION  OF  PAR- 
AXIAL  RAYS  AT  A  SPHERICAL  MIRROR. 

110.  In  the  accompanying  diagrams  (Figs.  52  and  53)  the  axis  of 
the  spherical  mirror  is  shown  by  the  straight  line  M  C.  The  straight 
line  MB  represents  an  incident  paraxial  ray  meeting  the  spherical  re- 
flecting surface  at  the  point  B.  In  Fig.  52  the  spherical  surface  is 
convex,  and  in  Fig.  53  it  is  concave.  At  the  point  B  the  ray  is  reflected 


rtp*E) 


FIG.  52. 
REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  MIRROR.    Convex  Mirror. 


AM=u,    AM'^u',    AC=r.    AF=r\2  =  —ft    FM=xt 
£  AMB  =  0,     L  AM'B  =  «'.     Z  BCA  =  4>.     L  NBM  =  £  WBN=  «. 


in  a  direction  BW,  such  that  Z  NBM  =  Z  WB  N,  where  BN  is  the 
normal  to  the  surface  at  the  point  B  drawn  in  the  medium  of  the  inci- 
dent and  reflected  rays.  Designating  by  M'  the  point  where  the  re- 
flected ray  crosses  the  axis,  either  really  (as  in  Fig.  53)  or  virtually 
(as  in  Fig.  52),  let  us  denote  by  the  symbols  u  and  u'  the  abscissae, 
with  respect  to  the  vertex  A  as  origin,  of  the  two  points  M  and  M' 
where  the  ray  crosses  the  axis  before  and  after  reflexion,  respectively; 
thus  AM  =  u,  AM'  =  u'.  Also,  as  in  §  108,  A  C  =  r. 

Since  the  normal  B  N  bisects  the  (interior  or  exterior)  angle  at  B 
of  the  triangle  MBM',  we  have: 

CM     M  'C. 
BM~~BM~f] 

and  since  the  point  B  is  very  close  to  A,  this  proportion  may  be 
written  : 

C_M  _  M'C 

AM~  AM'' 

where,  as  we  saw  above  (§  109),  magnitudes  of  the  second  order  of 


138 


Geometrical  Optics,  Chapter  V. 


[  §  no. 


smallness  are  neglected.1     Now 

CM  =  CA  +  AM  =  u-r,    M'C  =  M' A  +  AC  =  r  -  u' \ 
and,  therefore, 


u  —  r      r  —  u' 


or 


u 
I       I 


(61) 


Thus,  knowing  the  mirror  as  to  both  size  and  form  (which  means  that 
we  know  both  the  magnitude  and  sign  of  r),  and  being  given  the  posi- 
tion of  the  point  M  where  the  ray  crosses  the  axis  before  reflexion  at 
the  mirror,  we  can  determine  the  abscissa  of  the  point  M'  where  the 
ray  crosses  the  axis  after  reflexion. 

According  to  formula  (61),  any  paraxial  ray  which  crosses  the  axis 
before  reflexion  at  the  point  M,  will  cross  the  axis  after  reflexion  at 


w 


FIG.  53. 

REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  MIRROR.    Concave  Mirror. 
AM=*u,    AM'  =  u',    AC=r,    AF=tl2  =  —f.    FM  =  x,    FM'  =  x'  , 
L  AMB  =  9,     L  AAfB  =  0',     Z  BCA  =  <*>.     Z.  NBM  =  £  WBN=  a. 


the  point  M'  r.  Thus,  a  homocentric  bundle  of  paraxial  rays  incident 
on  a  spherical  mirror  remains  homocentric  after  reflexion  at  the  mirror. 
According  as  the  sign  of  the  abscissa  u'  is  positive  or  negative,  the 
point  M  '  will  lie  to  the  right  or  left  of  the  vertex  A  .  In  the  former 
case  the  image  at  M'  is  a  virtual  image  (Fig.  52),  whereas  in  the  latter 
case  we  have  a  real  image  at  M'  (Fig.  53)  ;  see  §  44. 

Moreover,  since  the  formula  is  symmetrical  with  respect  to  u  and  u'  , 
so  that  the  equation  remains  unaltered  when  we  interchange  the  letters 
u  and  u',  it  follows  that  if  M'  is  the  image  of  M,  then  M  will  also  be 

1  In  writing  this  proportion,  we  must  be,  careful  that  the  two  members  of  it  shall  have 
like  signs.  Thus,  in  the  diagrams,  as  here  drawn,  CM  and  AM  have  the  same  directions, 
so  that  for  these  diagrams  the  ratio  CM  I  AM  is  positive.  Hence,  if  the  ratio  M'C  I  AM' 
is  equal  to  CM  I  AM,  it  must  be  positive  also;  that  is,  M'C  and  AM'  must  likewise  have 
the  same  directions. 


§  111.]  Reflexion  and  Refraction  of  Paraxial  Rays.  139 

the  image  of  M'\  which  is  merely,  of  course,  an  illustration  of  the 
general  law  of  Optics  known  as  the  Principle  of  the  Reversibility  of  the 
Light- Path  (§  18).  But  the  symmetry  of  the  equation  implies  more 
than  is  involved  in  this  principle.  It  indicates  also  that,  in  the  case 
of  Reflexion,  Object-Space  and  Image-Space  coincide  completely:  the 
paths  of  the  incident  and  reflected  rays  both  lying  in  front  of  the 
mirror;  so  that  an  Object-Ray  and  an  Image-Ray  are  always  so  re- 
lated that  when  either  is  regarded  as  the  Object-Ray,  the  other  will 
be  the  corresponding  Image-Ray. 

The  magnitudes  denoted  by  u  and  u'  are  the  radii  of  the  incident 
and  reflected  wave-fronts  at  the  moment  when  the  disturbance  arrives 
at  the  vertex  A  of  the  mirror ;  and  hence  the  relation  given  by  formula 
(61)  may  also  be  expressed  as  follows:  The  algebraic  sum  of  the  curva- 
tures of  the  incident  and  reflected  waves  at  the  instant  when  the  disturb- 
ance arrives  at  the  vertex  of  the  spherical -mirror  is  egual  to  twice  the 
curvature  of  the  mirror. 

The  convergence  of  paraxial  rays  after  reflexion  (or  refraction)  at 
a  spherical  surface  is  said  to  be  a  "convergence  of  the  second  order"; 
which  means  that  the  second  and  higher  powers  of  the  incidence-angle 
a  are  neglected.  When  we  neglect  magnitudes  of  this  order,  the  spheri- 
cal surface  will  coincide  with  every  surface  of  revolution  which  has 
the  same  curvature  at  the  vertex;  so  that  the  formula  (61)  applies  to 
the  reflexion  of  paraxial  rays  at  a  surface  of  revolution  of  any  form, 
where  r  denotes  the  radius  of  curvature  at  the  vertex  of  a  meridian 
section  of  the  surface. 

111.     Since  CM  :AM  =  M' C  :AM',  it  follows  that 

CM     AM- 
CM':AM'~       If 

that  is,  the  anharmonic  or  double  ratio  of  the  four  axial  points  C,  A,  M 
and  M'  is 

(CAMM')  =  -  i;  (62) 

consequently,  the  points  C,  ,A ,  M,  M'  are  a  harmonic  range  of  points, 
the  Object-Point  M  and  its  Image-Point  M'  being  harmonically  sepa- 
rated by  the  centre  C  and  the  vertex  A  of  the  spherical  mirror.  Ac- 
cordingly, we  have  the  following  simple  construction  of  the  Image- 
Point  M'  due  to  the  reflexion  at  a  spherical  mirror  of  paraxial  rays 
emanating  originally  from  an  axial  Object-Point  M: 

On  any  straight  line,  supposed  to  represent  the  axis  of  the  spherical 
mirror,  take  three  points  A,  C,  M  (Fig.  54),  ranged  along  the  line  in 


140  Geometrical  Optics,  Chapter  V.  [  §  112. 

any  order  whatever;  the  letters  A  and  C  designating  the  positions 
of  the  vertex  and  centre,  respectively,  of  the  spherical  mirror,  and  the 
letter  M  designating  the  position  of  the  given  axial  Object-Point.  On 
any  other  straight  line  drawn  through  the  point  M  take  two  points  H 
and  J;  and  draw  CH  and  AJ  intersecting  in  a  point  P  and  A  H  and 
CJ  intersecting  in  a  point  Q.  The  straight  line  connecting  the  points 
P  and  Q  will  meet  the  axis  in  the  I  mage- Point  M'  conjugate  to  the 
Object-Point  M.  In  making  this  construction  a  straight-edge  is  the 

only  drawing-instrument 
that  will  be  needed.  The 
proof  of  the  construction 
is  obtained  at  once  from 
the  complete  quadrangle 
A  CHJ.  If  the  points  A 
and  C  in  the  diagram  are 
interchanged,  the  points 
M  and  M '  will  evidently 

REFLEXION    OF    PARAXIAL    RAYS    AT    A    SPHERICAL      be    a     pair    of    Conjugate 
MIRROR     Construction  of  Conjugate  Axial  Points  M.  M'.  {    ^  alsQ      ith  respect  to 

Centre  of  Mirror  at  C,  vertex  at  A.  ^ 

this  new  spherical  surface ; 

thus,  if  the  points  M,  M'  are  a  pair  of  axial  conjugate  points  with 
respect  to  a  spherical  mirror  with  its  centre  at  C  and  its  vertex  at 
Aj  these  same  points  will  be  conjugate  to  each  other  with  respect  to 
a  spherical  mirror  of  the  opposite  kind  with  its  centre  at  A  and  its 
vertex  at  C. 

112.  Focal  Point  and  Focal  Length  of  Spherical  Mirror.  In  the 
special  case  when  the  axial  Object-Point  M  coincides  with  the  infinitely 
distant  Object- Point  E  of  the  #-axis,  the  conjugate  point  M'  will  in 
this  case  be  situated  at  a  point  E'y  such  that 


that  is, 

AE'  =  E'C. 

Hence,  a  cylindrical  bundle  of  incident  paraxial  rays  parallel  to  the 
axis  will  be  transformed  by  reflexion  at  a  spherical  mirror  into  a  homo- 
centric  bundle  of  rays  with  its  vertex  at  a  point  E'  lying  midway 
between  the  vertex  and  centre  of  the  mirror. 

On  the  other  hand,  if  the  Image-Point  M'  coincides  with  the  in- 
finitely distant  Image-Point  F'  of  the  #-axis,  the  corresponding  Ob- 


§  112.]  Reflexion  and  Refraction  of  Paraxial  Rays.  141 

ject- Point  M  will  be  situated  on  the  axis  at  a  point  Fy  such  that 

(CAFF'}  =|J=  -is 

or 

AF  =  FC; 

that  is,  this  point  F,  which  is  the  vertex  of  a  bundle  of  incident  par- 
axial  rays  to  which  corresponds  a  cylindrical  bundle  of  reflected  rays 
all  parallel  to  the  axis,  is  also  situated  midway  between  the  vertex 
and  the  centre  of  the  spherical  mirror.  Thus,  in  the  case  of  a  spheri- 
cal mirror  the  two  points  F  and  Er  are  coincident. 

The  points  designated  by  F  and  Ef  are  called  the  Focal  Points  of 
the  optical  system. 

The  Focal  Length  of  a  Spherical  Mirror  may  be  defined  as  the 
abscissa  of  the  vertex  A  with  respect  to  the  Focal  Point  F]  thus,  if 
the  Focal  Length  is  denoted  by  the  symbol/,  we  have: 

FA=f=  -r/2. 

If  the  abscissae,  with  respect  to  the  Focal  Point  F,  of  the  conjugate 
axial  points  M,  M'  are  denoted  by  x,  x' ,  respectively;  that  is,  if  we  put 

FM  =  xt     FM'  =  x', 
then  we  have  at  once : 

«  =  *-/,  «'  =  *'-/, 

and  substituting  these  values  in  formula  (61),  we  obtain: 

xx'  =f;  (63) 

a  most  convenient  and  simple  form  of  the  abscissa-relation  of  conjugate 
axial  points,  which  contains  the  whole  theory  of  the  reflexion  of  par- 
axial  rays  at  a  spherical  mirror. 

According  as  the  Focal  Length  /  is  positive  or  negative,  the  mirror 
is  convex  or  concave.  Thus,  in  a  concave  mirror  the  Focal  Point  F 
lies  in  front  of  the  mirror,  so  that  incident  paraxial  rays  parallel  to  the 
axis  will  be  converged  by  reflexion  at  a  concave  mirror  to  a  real  focus 
at  F;  whereas  in  a  convex  mirror  the  Focal  Point  F  lies  beyond  the 
mirror  (to  the  right  of  the  vertex  A),  so  that  a  bundle  of  incident  par- 
axial  rays  which  are  parallel  to  the  axis  will  be  transformed  by  reflexion 
at  a  convex  mirror  into  a  bundle  of  rays  diverging  as  if  they  had  come 
from  a  virtual  focus  at  the  point  F. 

Whether  the  mirror  is  convex  or  concave,  and  whether  the  bundle 
of  incident  rays  is  convergent  or  divergent,  the  conjugate  axial  points  M, 


142  Geometrical  Optics,  Chapter  V.  [  §  113. 

M'  lie  always  on  the  same  side  of  the  Focal  Point  of  the  Spherical  Mirror; 
as  is  readily  seen  from  formula  (63). 

ART.  34.     EXTRA-AXIAL   CONJUGATE   POINTS  AND   THE   LATERAL   MAGNI- 
FICATION IN  THE  CASE  OF  THE  REFLEXION  OF  PARAXIAL 
RAYS  AT  A  SPHERICAL  MIRROR. 

113.     Graphical  Method  of  Showing  the  Imagery  by  Paraxial  Rays. 

Let  M,  M'  designate  the  positions  on  the  axis  of  a  spherical  mirror  of  a 
pair  of  conjugate  points,  constructed  according  to  the  method  given  in 
§  1 1 1 ;  and  connect  both  of  these  points  by  straight  lines  with  a  point 
V  on  the  surface  of  the  reflecting  sphere.  In  the  plane  of  these  lines 
draw  Ay  tangent  to  the  sphere  at  its  vertex  A,  and  let  B  and  G  desig- 
nate the  points  where  the  straight  lines  MV,  M'V  meet  the  straight 
line  Ay.  Also,  join  the  point  B  with  the  point  M'  by  a  straight  line. 
If  the  point  V  were  very  close  to  the  vertex  A ,  then  the  straight  line 
MV  would  be  the  path  of  an  incident  paraxial  ray  proceeding  from 
M,  and  the  path  of  the  corresponding  reflected  ray  would  be  VMf.  In 
this  case,  however,  the  points  designated  here  by  the  letters  V,  B  and 
G  would  all  be  so  near  together  that,  even  when  we  cannot  regard  V 
as  coincident  with  A,  we  can  regard  V,  B  and  G  as  coincident  with 
one  another;  and  therefore  we  might  take  the  straight  line  BMf  as 
representing  the  path  of  the  reflected  ray. 

In  the  construction  of  diagrams  exhibiting  the  procedure  of  paraxial 
rays  a  practical  difficulty  is  encountered  due  to  the  fact  that,  whereas 
in  reality  such  rays  are  comprised  within  the  very  narrow  cylindrical 
region  immediately  surrounding  the  axis  of  the  spherical  surface,  it  is 
obviously  impossible  to  show  them  in  this  way  in  a  figure,  because  we 
should  have  to  take  the  dimensions  of  the  figure  at  right  angles  to  the 
axis  so  small  that  magnitudes  of  the  second  order  of  smallness  in  such 
directions  would  no  longer  be  perceptible.  On  the  other  hand,  if  we 
were  to  represent  these  magnitudes  as  larger  than  they  actually  are, 
the  relations  which  we  have  found  above  would  no  longer  be  true  in 
the  case  of  such  lines;  thus,  for  example,  the  rays  in  the  drawing  would 
not  intersect  in  the  places  demanded  by  the  formulae. 

In  order  to  overcome  this  difficulty,  REUSCH  suggested  a  method  of 
drawing  these  diagrams  which  has  been  very  generally  adopted,  and 
which  in  large  measure  is  entirely  satisfactory.  Without  altering  the 
dimensions  parallel  to  the  axis,  the  dimensions  at  right  angles  to  the 
axis  are  all  magnified  in  the  same  proportion.  Thus,  for  example, 
if  the  ordinate  h  =  DB  (Fig.  51)  is  a  magnitude  of  the  order  i/k,  it  is 
shown  in  the  figure  magnified  k  times;  whereas  an  ordinate  of  magni- 


§  114.]  Reflexion  and  Refraction  of  Paraxial  Rays.  143 

tude  of  the  order  i  /k2,  that  is,  of  the  second  order  as  compared  with  h, 
even  in  the  magnified  diagram  would  be  shown  as  a  magnitude  of  the 
order  i/&;  so  that  if  k  is  infinite,  such  ordinates  as  h,  which  are  of  the 
first  order  of  smallness,  will  be  shown  in  the  figure  by  lines  of  finite 
length,  whereas  magnitudes  of  the  second  order  of  smallness  will  dis- 
appear completely  in  the  magnified  diagram. 

Of  course,  one  effect  of  this  lateral  enlargement  will  be  to  misrepre- 
sent to  some  extent  the  relations  of  the  lines  and  angles  in  the  figure. 
Thus,  for  example,  the  circle  in  which  the  spherical  surface  is  cut  by  the 
plane  of  a  meridian  section  will  thereby  be  transformed  into  an  infinitely 
elongated  ellipse  with  its  major  axis  perpendicular  to  the  axis  of  the 
spherical  surface,  that  is,  into  a  straight  line  Ay  tangent  to  the  circle 
at  its  vertex  A .  The  minor  axis  of  this  ellipse  remains  unchanged  and 
equal  to  the  diameter  of  the  circle,  and,  moreover,  the  centre  of  the 
ellipse  remains  at  the  centre  C  of  the  circle.  The  most  apparent 
change  will  be  in  the  angular  magnitudes  which  will  be  completely 
altered.  Thus,  for  example,  every  straight  line  drawn  through  the 
centre  C  really  meets  the  circle  normally,  but  in  the  distorted  figure 
the  axis  will  be  the  only  one  of  such  lines  which  meets  normally  the 
straight  line  which  takes  the  place  of  the  circle.  Angles  which  in 
reality  are  equal  will  appear  unequal,  and  vice  versa.  However — and 
this  after  all  is  the  really  essential  matter — the  relative  dimensions  of 
the  ordinates  and  the  absolute  dimensions  of  the  abscissae  will  not  be 
changed  at  all;  and,  therefore,  lines  which  are  really  straight  lines  will 
appear  as  straight  lines  in  the  figure,  and  straight  lines  which  are  really 
parallel  will  be  shown  in  the  figure  as  parallel  straight  lines.  The 
abscissa  of  the  point  of  intersection  of  a  pair  of  straight  lines  as  it 
appears  in  the  figure  will  be  the  real  abscissa  of  this  point. 

In  such  a  figure,  therefore,  any  ray,  no  matter  what  slope  it  may 
have,  nor  how  far  it  may  be  from  the  axis,  is  to  be  considered  as  a 
paraxial  ray.  The  meridian  section  of  the  spherical  surface  will  be 
represented  in  the  figure  by  the  straight  line  Ay  (the  y-axis),  and  the 
position  of  the  centre  C  with  respect  to  the  vertex  A  will  show  whether 
the  surface  is  convex  or  concave. 

114.  If  we  suppose  that  the  axis  of  the  spherical  mirror  is  rotated 
about  the  centre  C  through  a  very  small  angle  MCQ  so  that  the  axial 
point  M  moves  along  the  infinitely  small  arc  of  a  circle  to  a  point  Q, 
the  conjugate  axial  point  Mf  will  likewise  describe  an  infinitely  small 
arc  of  a  concentric  circle,  and  will  determine  a  point  Q'  on  the  straight 
line  joining  Q  with  C,  such  that  if  U  designates  the  point  where  the 
straight  line  QC  meets  the  spherical  surface,  the  points  Q,  Q'  will  be 


144 


Geometrical  Optics,  Chapter  V. 


§115. 


harmonically  separated  (§  in)  by  the  points  C,  U\  that  is,  (C  UQQ') 
=  —  i.  Thus,  the  point  Qf  is  evidently  the  image-point  conjugate 
to  the  Extra-Axial  Object-Point  Q.  If  the  Object-Points  lie  on  the 
element  of  a  spherical  surface  which  is  concentric  with  the  reflecting 
sphere,  the  corresponding  Image-Points  will  likewise  be  found  on  an 
element  of  another  concentric  spherical  surface,  and  any  straight  line 
going  through  the  centre  C  will  determine  by  its  intersections  with 
this  pair  of  concentric  surfaces,  of  radii  CM  and  CM' ',  a  pair  of  con- 
jugate points  such  as  Q,  Q'.  If,  as  we  assume  here,  the  angle  MCQ 
is  infinitely  small,  the  arcs  MQ,  M'Q'  may  be  regarded  as  very  short 
straight  lines  perpendicular  to  the  axis  at  M,  Mf,  respectively.  Ac- 
cordingly, on  the  supposition  that  the  only  rays  concerned  in  the  pro- 
duction of  the  image  are  such  rays  as  meet  the  reflecting  surface  at 
very  nearly  normal  incidence,  the  following  conclusions  may  be  drawn: 
(i)  The  image,  in  a  spherical  mirror,  of  a  plane  object  perpendicular 
to  the  axis  is  likewise  a  plane  perpendicular  to  the  axis;  (2)  A  straight 
line  passing  through  the  centre  of  the  spherical  mirror  intersects  a  pair  of 
such  conjugate  planes  in  a  pair  of  conjugate  points;  and  (3)  To  a  homo- 
centric  bundle  of  incident  paraxial  rays  proceeding  from  a  point  Q  in  a 

plane  perpendicular  to  the  axis 
of  the  spherical  mirror  there 
corresponds  a  homocentric  bun- 
dle of  reflected  rays  with  its 
vertex  Q'  lying  in  the  conjugate 
Image- Plane. 

115.  In  order  to  construct 
the  Image- Point  Q'  of  the 
Extra- Axial  Object- Point  Q, 
we  have  merely  to  find  the 
point  of  intersection  after  re- 
flexion at  the  spherical  mirror 
of  any  two  rays  emanating 
originally  from  the  point  Q. 
The  two  diagrams  (Figs.  55 
and  56),  which  are  drawn  ac- 
cording to  the  method  de- 
scribed above  (§  113),  exhibit  this  construction  in  the  case  of  both  a 
concave  and  a  convex  mirror.  Of  the  incident  rays  proceeding  from 
Q  it  is  convenient  to  select  the  following  pair  for  this  construction: 
the  incident  ray  QC  which  proceeding  towards  the  centre  of  the 
mirror  C  meets  the  spherical  surface  normally  at  the  point  £7,  whence 


FIG.  55  and  FIG.  56. 

REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL 
MIRROR.  Construction  of  point  Q*  conjugate  to 
extra-axial  Object-Point  Q.  In  the  diagrams  the 
meridian  section  of  the  mirror  is  represented  by  a 
straight  line  Ay  perpendicular  to  the  axis  of  the 
mirror  at  the  vertex  A.  The  straight  line  M'Q?  per- 
pendicular to  the  axis  is  the  image  of  the  straight 
line  MQ  also  perpendicular  to  the  axis. 


§  116.]  Reflexion  and  Refraction  of  Paraxial  Rays.  145 

it  is  reflected  back  along  the  same  path,  and  the  incident  ray  QV 
which  proceeding  parallel  to  the  axis  and  meeting  the  mirror  in  the 
point  designated  by  V  is  reflected  at  V  along  the  straight  line  joining 
Fwith  the  Focal  Point  F.  The  Image-Point  Qf  will  be  the  point  of 
intersection  of  this  pair  of  reflected  rays.  Moreover,  having  located 
the  position  of  Q',  we  can  draw  QM  and  Q'M'  perpendicular  to  the 
axis  at  M  and  M',  respectively;  then  M'Q'  will  be  the  image  of  the 
straight  line  MQ  perpendicular  to  the  axis  at  M.  In  Fig.  55  the  case 
is  shown  where  the  image  M'Q'  is  real  and  inverted;  whereas  in  Fig. 
56  the  image  M'Q'  is  virtual  and  erect.  Whether  the  image  is  real  or 
virtual  and  erect  or  inverted  will  depend  on  the  position  of  the  object 
with  respect  to  the  mirror  as  well  as  on  whether  the  mirror  is  convex 
or  concave. 

116.  The  Lateral  Magnification.  If  the  ordinates  of  the  pair  of 
extra-axial  conjugate  points  Q,  Qr  are  denoted  by  y,  y'  ,  respectively, 
that  is,  if  MQ  =  y,  M'Q'  =  /,  the  ratio  y'/y  is  called  the  Lateral 
Magnification  at  the  axial  point  M.  This  ratio  will  be  denoted  by  F; 
thus,  F  =  y'  fy.  The  sign  of  this  function  Y  indicates  whether  the 
image  is  erect  or  inverted;  if  F  is  positive,  as  in  Fig.  56,  the  image 
will  be  erect;  whereas  if  Y  is  negative,  as  in  Fig.  55,  the  image  will 
be  inverted.  The  absolute  value  of  Y  depends  on  the  relative  heights 
of  the  object  and  its  image;  it  will  be  greater  than,  equal  to,  or  less 
than,  unity,  according  as  the  height  of  the  image  is  greater  than,  equal 
to,  or  less  than,  that  of  the  object.  A  very  simple  investigation  shows 
how  F  is  a  function  of  the  abscissa  of  the  axial  point  M.  Since  the 
triangles  MCQ,  M'CQ'  (Figs.  55  and  56)  are  similar, 

M'Q':MQ  =  M'C:MC\ 
and  since 


and,  by  formula  (61),  we  have: 


r  —  u  u 

we  derive  the  following  formula  for  the  Lateral  Magnification  in  the 
case  of  a  Spherical  Mirror: 

r-£--^  (64) 

y  u 

Or,  in  case  we  wish  to  obtain  F  as  a  function  of  the  abscissa  x 
11 


146  Geometrical  Optics,  Chapter  V. 

(x  =  FM),  we  obtain  from  the  diagrams  directly: 


[§117. 


M'Q'      M'Q' 
MQ  =:   AV     ~~  FA  ' 

and  putting  FM'  =  x',  FA  =/,  and  using  also  formula  (63),  we  have: 

F=?  =  *  =  7:  (6s) 

which  of  course  is  likewise  easily  deducible  from  (64). 

Either  of  the  two  pair  of  formulae  (61)  and  (64)  or  (63)  and  (65) 
determine  completely  the  Imagery  in  the  case  of  the  Reflexion  of 
Paraxial  Rays  at  a  Spherical  Mirror. 

117.  If  the  axial  Object-Point  M  is  supposed  to  travel  along  the 
axis  of  the  spherical  mirror,  and  if  at  the  same  time  the  point  Q  is 
supposed  to  travel  with  an  equal  velocity  along  a  line  parallel  to  the 
axis,  the  corresponding  manoeuvres  of  the  image  M'Q'  will  be  easily 
perceived  by  an  inspection  of  the  diagram  (Fig.  57),  which  shows  the 


0iy*et  ffay 


FIG.  57. 

REFLEXION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  MIRROR.  The  numerals  1,  2,  3,  etc.,  ranged 
from  left  to  right  along  a  straight  line  parallel  to  the  axis  of  the  mirror  indicate  the  successive 
positions  of  an  object-point,  and  the  numerals  1',  2',  3',  etc.,  show  the  corresponding  positions  of 
the  image-point  ranged  along  the  straight  line  VF.  The  case  shown  in  the  figure  is  for  a  Concave 
Mirror.  The  straight  lines  11',  22',  33',  etc..  all  intersect  at  the  centre  C  of  the  mirror.  If  the 
object-point  is  virtual  (as  at  7  or  8),  the  image  in  a  concave  mirror  will  be  real. 

case  of  a  concave  mirror  with  its  Focal  Point  F  in  front  of  the  mirror. 
Let  us  suppose  that  the  Object  moves  from  left  to  right  starting  from 
an  infinite  distance  in  front  of  the  mirror.  The  numerals  I,  2,  3,  etc., 
are  used  to  designate  a  number  of  successive  positions  of  the  Object- 
Point  Q,  whereas  the  same  numerals  with  primes  show  the  correspond- 


§  118.]  k  Reflexion  and  Refraction  of  Paraxial  Rays.  147 

ing  positions  of  the  Image-Point  Q '.  Evidently,  all  the  straight  lines 
n',  22',  33',  etc.,  will  pass  through  the  centre  C  of  the  mirror.  So 
long  as  the  Object  MQ  lies  in  front  of  the  Focal  Point  F  the  image 
M'Q'  in  the  concave  mirror  is  real  and  inverted.  As  MQ  advances 
towards  the  centre  C,  the  Image  M'Q'  proceeds  between  F  and  C  also 
towards  the  centre  C,  and  Object  and  Image  arrive  together  in  the 
plane  perpendicular  to  the  axis  at  C,  the  Image  being  then  of  the  same 
size  as  the  Object,  but  inverted.  As  the  Object  proceeds  past  C 
towards  F,  the  real  and  inverted  Image  proceeds  in  the  opposite  direc- 
tion towards  infinity;  so  that  when  the  point  Q  arrives  at  the  point 
marked  4  in  the  Focal  Plane,  the  point  Qf  is  the  infinitely  distant  point 
of  the  straight  line  VF.  As  the  Object  continues  its  journey  from  the 
Focal  Point  F  towards  the  vertex  A  of  the  mirror,  the  Image,  which  is 
now  virtual  and  erect,  travels  from  infinity  towards  the  vertex  A, 
and  Object  and  Image  arrive  together  at  the  vertex  and  coincide  with 
each  other  there.  If  the  Object  proceeds  beyond  the  vertex,  we  shall 
have  then  the  case  of  a  virtual  Object,  to  which  there  corresponds  a  real 
erect  Image  lying  between  the  vertex  A  and  the  Focal  Point  F.  The 
Image,  it  will  be  observed,  travels  always  in  a  direction  opposite  to 
that  taken  by  the  Object;  which  is  a  characteristic  property  of  re- 
flexion. Moreover,  it  will  be  noted  that  Object  and  Image  lie  always 
on  the  same  side  of  the  Focal  Plane. 

II.  REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE. 

ART.  35.     CONJUGATE  AXIAL  POINTS  IN  THE  CASE  OF  THE  REFRACTION 
OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE. 

118.  In  the  diagrams  (Figs.  58  and  59)  the  plane  of  the  paper  repre- 
sents the  meridian  section  of  a  spherical  refracting  surface  separating 
two  isotropic  optical  media  of  absolute  refractive  indices  n  and  n' . 
In  Fig.  58  the  centre  C  lies  in  the  second  medium  (n'),  so  that  the 
spherical  surface  is  convex;  whereas  in  Fig.  59  the  centre  C  lies  in 
the  first  medium  (n),  and  the  spherical  surface  is  concave.  The  axis 
of  the  refracting  sphere  is  the  straight  line  xx  which  joins  the  centre  C 
with  the  vertex  A.  The  letters  in  these  figures  have  the  same  mean- 
ings as  in  the  corresponding  diagrams  for  the  reflexion  of  paraxial 
rays  at  a  spherical  mirror. 

An  incident  ray  meeting  the  spherical  surface  will  be  refracted  in 
a  direction  such  that,  if  a  and  a'  denote  the  angles  of  incidence  and 
refraction,  then,  by  the  Law  of  Refraction: 


148 


Geometrical  Optics,  Chapter  V. 


[  §  118. 


If  M,  Mf  designate  the  points  where  the  ray  crosses  the  axis  before  and 
after  refraction  at  the  spherical  surface,  and  if  BN  is  the  normal  to 


FIG.  58  and  FIG.  59. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE  SEPARATING  TWO  MEDIA  OF 
INDICES  n,  n'  . 

AM=u,    AM'  =  u',    AC=r,    FA=f,    E'A=S,    FM=x, 


the  spherical  refracting  surface  at  the  incidence-point  B  drawn  from 
B  into  the  first  medium,  then  in  the  figures: 


also  if  <p  denotes  the  angle  subtended  at  the  centre  C  by  the  arc  AB, 
then,  according  to  the  definition  of  <p  given  in  §  108,  ABC  A  =  <p. 
In  the  triangles  MBC  and  M'BC  we  have: 

CM  :  BM  =  sin  a  :  sin  <f>,     CM'  :  BM'  =  sin  a'  :  sin  ^>, 
and,  hence,  dividing  one  of  these  equations  by  the  other,  we  obtain: 

CM    BM  _rf 
CM'  ''  BM'  ~  n  ' 

Since  the  incidence-point  B  is  supposed  to  be  so  near  A  that  we  can 
neglect  magnitudes  of  the  second  order  of  smallness,  we  may  write  A 
in  this  equation  in  place  of  B;  and  thus  we  obtain  for  the  refraction 
of  a  paraxial  ray  at  a  spherical  surface: 

CM_    AM      n' 
CM'1  AM'~  n' 


SJJFORgjj 
§  119.]  Reflexion  and  Refraction  of  Paraxial  Rays.  149 

or 

(CAMM')=  £;  (66) 

that  is,  The  Double  (or  Anharmonic)  Ratio  of  the  four  axial  points  C,  A, 
Mt  M'  is  constant,  and  equal  to  the  relative  index  of  refraction  of  the  two 
media. 

Thus,  for  a  given  spherical  refracting  surface,  the  axial  point  Mf 
corresponding  to  a  given  axial  point  M  is  a  perfectly  definite  point, 
and  accordingly  we  derive  the  following  result: 

To  a  homocentric  bundle  of  incident  paraxial  rays  with  its  vertex  lying 
on  the  axis  of  the  spherical  refracting  surface  there  corresponds  also  a 
homocentric  bundle  of  refracted  rays  with  its  vertex  lying  on  the  axis. 

Thus,  if  M  designates  the  position  of  an  axial  Object-Point,  its 
image  produced  by  the  refraction  of  paraxial  rays  at  a  spherical  sur- 
face will  be  at  a  point  M'  on  the  axis.  In  Fig.  58  we  have  at  M'  a  real 
image  of  the  Object-Point  M ;  whereas  in  Fig.  59  the  image  is  virtual. 
The  four  points  M,  M',  A  and  C  may  be  ranged  along  the  axis  in  any 
order  whatever,  depending  on  the  form  of  the  spherical  refracting 
surface  and  on  whether  n  is  greater  or  less  than  n'.  If  the  incident 
rays  converge  towards  a  point  M  lying  on  the  axis  beyond  (or  to  the 
right  of)  the  vertex  A,  the  point  M  will  be  a  virtual  Object- Point; 
but  in  this  case,  as  in  all  cases,  the  corresponding  Image-Point  M'  can 
be  found  by  formula  (66). 

Moreover,  if  (CAMM1)  =  n'/n,  then  also  (CAM'M)  =  njn'. 
Thus,  if  a  ray  proceeding  from  an  axial  point  M  in  the  first  medium 
crosses  the  axis  after  refraction  at  the  spherical  surface  at  the  point 
M'  in  the  second  medium  (see  §  10),  then  also  a  ray  proceeding  from 
the  point  M'  in  the  second  medium  will  be  refracted  at  the  spherical 
surface  so  as  to  cross  the  axis  at  the  point  M.  This  is  in  accordance 
with  the  general  Principle  of  the  Reversibility  of  the  Light- Path  (§  18). 
If  M'  is  the  image  of  M,  M  will  be  likewise  the  image  of  M' . 

119.  Construction  of  the  Image-Point  M'  conjugate  to  the  Axial 
Object-Point  M.  The  following  is  a  simple  method  of  constructing 
the  Image-Point  M'  corresponding  to  an  Object-Point  M  lying  on  the 
axis  of  a  spherical  refracting  surface.  Through  the  centre  C  draw  any 
straight  line,  and  take  on  it  two  points  so  situated  that  their  distances 
from  C  are  in  the  ratio  n'  :  n.  Instead  of  drawing  any  straight  line 
through  C,  it  will  be  convenient  to  take  this  line  perpendicular  to  the 
axis  at  C,  as  is  done  in  Fig.  60.  Let  G,  G'  designate  the  positions  on 
this  line  of  two  points  whose  distances  from  C  are  such  that  we  have : 


150 


Geometrical  Optics,  Chapter  V. 


[  §  120. 


CG  :  CGr  =  n'  :n.  Through  the  vertex  A  of  the  spherical  refracting 
surface  draw  the  straight  line  Ay  parallel  to  the  straight  line  drawn 
through  C;  if  this  latter  is  perpendicular  to  the  axis,  the  straight  line 


FIG.  60. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.  Construction  of  axial  point  Mf 
conjugate  to  axial  Object-point  M.  C  is  centre,  A  is  vertex  and  xx  axis  of  Spherical  Refracting 
Surface.  CG'  GTis  perpendicular  to  axis ;  CG :  CG'  =  n'  :n.  Tis  infinitely  distant  point  of  jy-axis. 

Ay  will  be  tangent  to  the  spherical  surface  at  the  point  A.  Join  the 
axial  Object- Point  M  with  the  point  G  by  a  straight  line,  and  let  Ym 
designate  the  point  where  this  line  meets  Ay,  then  the  straight  line 
joining  the  points  Ym  and  G'  will  meet  the  axis  in  the  required  point  M'. 
For,  evidently,  the  point  Ym  is  the  centre  of  perspective  of  the  two 
projective  point-ranges  C,  A,  M,  M'  and  C,  T,  G,  G',  where  T  desig- 
nates the  infinitely  distant  point  of  the  straight  line  which  intersects 
the  axis  at  C;  and  since,  by  construction, 

CG  .  TG  _  CG  _  n' 
~CG'  •  TG'  ~  CG'  ~  n  ' 
it  follows  that  we  must  have: 

(CAMM')  =  — , 

in  accordance  with  formula  (66). 

120.  The  Focal  Points  F  and  E'  of  a  Spherical  Refracting  Surface. 
Evidently,  the  vertex  A  and  the  centre  C  of  the  spherical  refracting 
surface  are  two  self -corresponding  points  of  the  two  projective  ranges 
of  Object-Points  and  Image-Points  lying  along  the  axis.  Let  us  dis- 
tinguish these  two  ranges  of  corresponding  points  by  the  letters  x  and 
#',  and  let  E  and  F'  designate  the  infinitely  distant  points  of  x  and  x', 
respectively.  Thus  E  is  the  infinitely  distant  axial  Object-Point  and 
F'  is  the  infinitely  distant  axial  Image-Point.  In  order  to  find  the 
Image-Point  E'  conjugate  to  the  infinitely  distant  Object-Point  E, 


§  120.]  Reflexion  and  Refraction  of  Paraxial  Rays.  151 

we  must  draw  through  the  point  G  (Fig.  60)  a  straight  line  parallel  to 
the  axis  meeting  the  straight  line  A  y  in  the  point  designated  by  Ye; 
and  then  the  straight  line  YeGf  will  determine  by  its  intersection  with 
the  axis  the  required  point  £'.  Similarly,  in  order  to  find  the  position 
on  the  axis  of  the  Object-Point  F  corresponding  to  the  infinitely  dis- 
tant Image-Point  Ff,  we  must  draw  through  the  point  Gf  a  straight 
line  parallel  to  the  axis  and  meeting  the  straight  line  Ay  in  a  point  Yf\ 
and  then  the  straight  line  GYf  will  determine  by  its  intersection  with 
the  axis  the  required  point  F. 

Thus,  a  paraxial  ray  which  before  refraction  is  parallel  to  the  axis 
of  the  spherical  surface  will,  after  refraction,  cross  the  axis  (really  or 
virtually)  at  the  point  designated  by  £';  and,  also,  a  paraxial  ray 
which  before  refraction  crosses  the  axis  (really  or  virtually)  at  the 
point  designated  by  F  will,  after  refraction,  be  parallel  to  the  axis. 
These  points  F  and  Ef  are  called  the  Focal  Points;  the  point  F  is 
called  the  Focal-Point  of  the  Object-Space  or  the  Primary  Focal  Point, 
and  the  point  Ef  is  called  the  Focal  Point  of  the  Image-Space  or  the 
Secondary  Focal  Point.  The  two  Focal  Points  of  an  optical  system 
are  always  of  the  highest  importance. 

A  mere  inspection  of  the  diagram  (Fig.  60)  shows  that  the  Focal 
Points  F  and  Ef  of  a  spherical  refracting  surface  are  situated  so  that 

FA  =  CE',    E'A  =  CF\  (67) 

and,  hence,  we  have  the  following  rule: 

The  Focal  Points  of  a  Spherical  Refracting  Surface  are  so  situated 
on  the  axis  that  the  step  from  one  of  them  to  the  vertex  A  is  identical 
with  the  step  from  the  centre  C  to  the  other  one. 

This  result  may  also  be  stated  in  a  different  way;  for,  since 

FA  =  CE'  =  CA  +  AE', 
we  have  also  the  following  relation: 

AF+AE'  =  AC-,  (68) 

that  is,  The  algebraic  sum  of  the  distances  of  the  Focal  Points  from  the 
vertex  of  the  spherical  refracting  surface  is  always  equal  to  the  distance  of 
the  centre  from  the  vertex. 

Another  useful  relation,  obtained  from  the  two  similar  triangles 
AYf  F  and  AYe  Ef  is  the  proportion: 


AEf      CG 


152 


Geometrical  Optics,  Chapter  V. 


[  §  120. 


or 


AE> 
AF 


(69) 


which  may  be  put  in  words  as  follows :  The  two  Focal  Points  F  and  Ef 
of  a  spherical  refracting  surface  lie  on  opposite  sides  of  the  vertex,  and  at 
distances  from  it  which  are  in  the  ratio  n  :  n' . 

The  answer  to  the  question,  Which  of  the  two  Focal  Points  lies  in 
the  first  medium,  and  which  in  the  second  medium?  will  depend  on 
each  of  two  things,  viz.:  (i)  Whether  the  spherical  surface  is  convex 
or  concave,  and  (2)  Whether  n'  is  greater  or  less  than  n.  Thus,  for 
example,  if  the  rays  are  refracted  from  air  to  glass  (n' /n  =  3/2),  we 
find  from  formulae  (68)  and  (69)  AF  =  zCA,  AEf  =  3 AC',  so  that, 
starting  at  the  vertex  A  and  taking  the  step  CA  twice,  we  can  locate 
the  Primary  Focal  Point  F,  and  returning  to  the  vertex  A  and  taking 


FIG.  61. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.  Construction  of  the  Focal  Points 
.Fand  Et.  In  I  and  II  the  rays  are  refracted  from  air  to  glass.  In  III  and  IV  the  rays  are  refracted 
from  glass  to  air.  In  I  and  III  spherical  refracting  surface  is  convex.  In  II  and  IV  spherical 
refracting  surface  is  concave.  In  I  and  IV  incident  rays  parallel  to  the  axis  are  converged  to  a 
real  focus  at  Er ;  whereas  in  II  and  III  E'  is  a  virtual  focus. 

the  step  A  C  three  times,  we  can  locate  the  Secondary  Focal  Point  E'. 
The  two  diagrams  I  and  II  (Fig.  61)  show  the  positions  of  the  Focal 
Points  in  the  case  when  the  rays  are  refracted  from  air  to  glass  at  a 
convex  and  at  a  concave  spherical  surface.  It  will  be  seen  that  for 
this  case  the  Primary  Focal  Point  of  the  concave  surface  lies  in  the 
second  medium  (virtual  focus),  whereas  the  Primary  Focal  Point  of 
the  convex  surface  lies  in  the  first  medium  (real  focus).  On  the  other 
hand,  in  case  the  rays  are  refracted  from  glass  to  air  (n' In  =  2/3), 
we  have  AF  =  3 A  C,  AE'  =  2CA,  and  now  the  Primary  Focal  Point 
of  a  convex  spherical  refracting  surface  will  lie  in  the  second  medium 
and  the  Primary  Focal  Point  of  a  concave  surface  will  lie  in  the  first 
medium,  as  is  shown  in  the  diagrams  III  and  IV  (Fig.  61). 


§  122.]  Reflexion  and  Refraction  of  Paraxial  Rays.  153 

ART.  36.     REFRACTION    OF    PARAXIAL    RAYS    AT    A    SPHERICAL    SURFACE. 

EXTRA-AXIAL   CONJUGATE   POINTS.     CONJUGATE   PLANES.     THE 

FOCAL   PLANES   AND    THE    FOCAL   LENGTHS. 

121.  To  an  Object- Point  Q  lying  not  on  the  axis,  but  very  near  to 
it,  evidently  there  will  correspond  an  Image-Point  Q'  lying  on  the 
straight  line  joining  Q  with  the  centre   C  of  the  spherical  refracting 
surface,  the  position  of  which  is  determined  by  the  equation 

(CUQQ')  =  n'/n, 

where  U  designates  the  point  where  the  self -corresponding  ray  QQ' 
meets  the  spherical  surface.  Employing  here  exactly  the  same  reason- 
ing as  was  used  in  §  114  in  the  similar  case  of  Reflexion  at  a  Spherical 
Mirror,  we  may  copy  verbatim  the  results  which  were  obtained  there, 
merely  changing  the  words  "mirror",  "reflexion",  etc.,  to  adapt  the 
statements  to  the  case  of  refraction  at  a  spherical  surface.  Thus: 
(i)  The  image  of  a  plane  object  perpendicular  to  the  axis  of  a  spherical 
refracting  surface  is  likewise  a  plane  perpendicular  to  the  axis;  (2)  A 
straight  line  drawn  through  the  centre  of  the  spherical  refracting  surface 
will  intersect  a  pair  of  such  conjugate  planes  in  a  pair  of  conjugate  points; 
and  (3)  To  a  homocentric  bundle  of  incident  paraxial  rays  proceeding 
from  a  point  Q  in  a  plane  perpendicular  to  the  axis  of  the  spherical 
refracting  surface  there  corresponds  a  homocentric  bundle  of  refracted  rays 
with  its  vertex  Qf  lying  in  the  conjugate  Image- Plane. 

122.  The  Construction  of  the  Image-Point  Q'  Corresponding  to  the 
Extra-Axial  Object-Point  Q  may  be  performed  also  by  a  process  pre- 
cisely similar  to  that  used  in  §  115.     Thus,  in  the  diagrams  (Figs.  62 
and  63),  which  are  drawn  according  to  the  plan  explained  in  §  113,  the 
points  designated  by  the  letters  A  and   C  represent  the  vertex  and 
centre,  respectively,  of  the  spherical  refracting  surface.     In  Fig.  62 
the  surface  is  convex,  and  in  Fig.  63  it  is  concave.     If  the  positions 
of  the  Focal  Points  F  and  E'  are  not  assigned,  they  can  be  determined 
directly  by  the  relations  given  in  formulae  (68)  and  (69).     Both  of  the 
diagrams  show  the  case  when  n'  is  greater  than  n. 

The  incident  ray  proceeding  from  the  point  Q  towards  the  centre 
C  will  meet  the  spherical  surface  normally  and  will  continue  its  route 
into  the  second  medium  without  change  of  direction.  Thus,  as  was 
stated  also  above,  the  corresponding  point  Qf  must  lie  on  the  straight 
line  joining  Q  with  C.  To  the  incident  ray  QV  proceeding  from  the 
Object-Point  Q  parallel  to  the  axis  and  meeting  the  straight  line  Ay 
in  the  point  V  there  corresponds  a  refracted  ray  which  passes  (really 
or  virtually)  through  the  secondary  Focal  Point  E'.  Thus,  the  Image- 


154 


Geometrical  Optics,  Chapter  V. 


[  §  123. 


Point  Qf  will  be  at  the  point  of  intersection  of   the  straight  lines  QC 
and  VE'. 

The  intersection  of  any  pair  of  refracted  rays  emanating  originally 
from  the  Object- Point  Q  will  determine  the  position  of  the  Image- 
Point  Qf.  Thus,  for  example,  instead  of  one  of  those  used  above,  we 


FIG.  62  and  FIG.  63. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.  Construction  of  Image-Point  Cf 
corresponding  to  extra-axial  Object-Point  Q.  The  points  A  and  C  designate  the  positions  of  the 
vertex  and  centre  of  the  spherical  refracting  surface,  and  .Fand  E'  designate  the  positions  of  the 
Focal  Points.  In  Fig.  62  the  surface  is  convex,  in  Fig.  63  it  is  concave ;  for  both  diagrams  n'  >  n. 
In  Fig.  62  M'Q?  is  a  real,  inverted  image  of  MQ;  whereas  in  Fig.  63  the  image  is  virtual  and  erect. 

might  have  employed  the  ray  which  proceeding  from  the  Object- Point 
Q  towards  the  primary  Focal  Point  F  and  meeting  the  straight  line  Ay 
in  the  point  designated  in  the  diagrams  by  W  is  refracted  parallel  to 
the  axis  of  the  spherical  surface. 

If  M,  Mf  designate  the  feet  of  the  perpendiculars  let  fall  from  <2,  Q', 
respectively,  on  the  axis,  then  M'Q'  will  be  the  image,  by  paraxial 
rays,  of  the  infinitely  small  straight  line  MQ.  In  Fig.  62  this  image 
is  real  and  inverted,  whereas  in  Fig.  63  it  is  virtual  and  erect. 

123.  The  Focal  Planes  of  a  Spherical  Refracting  Surface.  If  the 
Object- Point  Q  is  the  infinitely  distant  point  of  the  straight  line  Q  C 
(Fig.  64),  it  will  be  a  point  of  the  infinitely  distant  plane  of  the  Object- 
Space  to  which  is  conjugate  a  plane  perpendicular  to  the  axis  at  the 
Focal  Point  E'  of  the  Image-Space.  This  plane  is  called  the  Focal 
Plane  of  the  Image-Space  or  the  Secondary  Focal  Plane.  Its  trace  in 
the  plane  of  the  paper  (which  shows  a  meridian  section  of  the  spherical 
surface)  is  the  straight  line  e'  which  we  may  call  the  secondary  Focal 
Line.  Thus,  we  can  say: 

To  a  bundle  of  parallel  incident  paraxial  rays  there  corresponds  a 


§  124.] 


Reflexion  and  Refraction  of  Paraxial  Rays. 


155 


homocentric  bundle  of  refracted  rays  with  its  vertex  lying  in  the  secondary 
focal  plane  of  the  spherical  refracting  surface. 

Similarly,  the  plane  perpendicular  to  the  axis  at  the  primary  Focal 
Point  F  is  called  the  Focal  Plane  of  the  Object-Space  or  the  Primary 


FIG.  64  (a). 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.  Incident  Parallel  Rays  intersect 
after  refraction  in  a  point  of  the  focal  plane  of  the  Image-Space,  the  trace  of  which  in  the  plane  of 
the  paper  is  the  focal  line  e! '. 

Focal  Plane,  and  its  trace  in  the  plane  of  the  paper  (Fig.  64)  is  the 
straight  line  /,  which  we  may  call  the  Primary  Focal  Line  in  the  plane 
of  this  meridian  section.  The  Image-Plane  conjugate  to  the  Primary 
Focal  Plane  is  the  infinitely  distant  plane  of  the  Image-Space;  and, 
hence,  if  the  Object- Point  Q  lies  in  the  Primary  Focal  Plane,  the  cor- 


FIG.  64 


REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.  Incident  Rays  emanating  from  a 
point  of  the  Focal  Plane  of  the  Object-Space  (the  trace  of  which  in  the  plane  of  the  paper  is  the 
Focal  I,ine  /")  are  made  parallel  by  refraction. 

responding  Image-Point  Qf  will  be  the  infinitely  distant  point  of  the 
straight  line  QC.  Thus: 

To  a  homocentric  bundle  of  incident  par  axial  rays,  with  its  vertex 
lying  in  the  Primary  Focal  Plane  of  the  spherical  refracting  surface, 
there  corresponds  a  bundle  of  parallel  refracted  rays. 

124.  The  Focal  Lengths  /  and  e'  of  a  Spherical  Refracting  Surface. 
In  Fig.  65  the  points  designated  by  M,  M'  are  the  points  where  a 
paraxial  ray  crosses  the  axis,  before  and  after  refraction,  respectively, 


156 


Geometrical  Optics,  Chapter  V. 


[  §  124. 


at  a  spherical  surface,  and  the  point  B  is  the  incidence-point  of  this 
ray.  The  vertex  of  the  spherical  surface  is  at  the  point  marked  A, 
and  the  Focal  Points  are  at  F  and  Ef.  Let 


where  6,  6'  denote  the  slope-angles  (§  108)  of  the  ray  before  and  after 
refraction,  respectively.  Through  the  Primary  Focal  Point  F  draw 
FK'  parallel  to  the  incident  ray  MB  and  meeting  the  straight  line 
Ay  in  the  point  designated  by  Kf,  and  through  the  Secondary  Focal 
Point  Ef  draw  GE'  parallel  to  the  refracted  ray  BM'  and  meeting  Ay 


FIG.  65. 

REFRACTION  OF  PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE.    The  focal  lengths  of  the  spherical 
refracting  surface  are : 


FA,    /  = 


tan*' 


••E'A, 


where  FR  =  g,  E?S'  =  tf  denote  the  intercepts  on  focal  planes  of  incident  ray  MB  and  correspond- 
ing refracted  ray  BM' ,  and  L  AMB  =  6,  £  AM' B  =  9f  are  the  slopes  of  incident  and  refracted  rays» 

in  the  point  designated  by  G.  Through  the  points  G  and  Kr  draw 
straight  lines  parallel  to  the  axis  of  the  spherical  refracting  surface; 
the  former  meeting  the  incident  ray  MB  in  the  point  designated  by 
R,  which  is  the  Object-Point  corresponding  to  the  infinitely  distant 
Image-Point  R'  of  the  refracted  ray  BM'  \  and  the  latter  meeting  the 
refracted  ray  BM'  in  the  point  S' ',  which  is  the  Image-Point  corre- 
sponding to  the  infinitely  distant  Object-Point  S  of  the  incident  ray 
MB.  The  point  R  will  lie  in  the  Primary  Focal  Plane,  and  the  point 
S'  will  lie  in  the  Secondary  Focal  Plane.  Let  us  put 

FR  =  AG  =  g,    E'S'  =  AK'  =  V. 
Evidently,  we  have  then  the  following  relations: 
£  k' 


tan  0' 


tan  0 


so  that  whatever  be  the  slopes  of  the  incident  and  refracted  rays,  the 


§  124.]  Reflexion  and  Refraction  of  Paraxial  Rays.  157 

intercepts  g  and  kr  will  always  be  such  that  the  above  ratios  have 
constant  values.  If  we  denote  these  constant  values  by  /  and  e',  that 
is,  if  we  put 

FA=f,     E'A=ef, 

these  equations  can  be  written  : 


The  constants  denoted  here  by  the  symbols  /  and  e'  are  called  the 
Primary  and  Secondary  Focal  Lengths,  respectively,  of  the  spherical 
refracting  surface.  The  proper  definitions  of  the  Focal  Lengths  (see 
§  178)  are  given  by  formulae  (70);  thus: 

The  Primary  Focal  Length  (f)  is  equal  to  the  Quotient  of  the  distance 
from  the  optical  axis  of  the  point  where  a  refracted  ray  crosses  the  Secondary 
Focal  Plane  by  the  tangent  of  the  slope-angle  of  the  corresponding  inci- 
dent ray;  and  similarly: 

The  Secondary  Focal  Length  (ef)  is  equal  to  the  quotient  of  the  distance 
from  the  optical  axis  of  the  point  where  an  incident  ray  crosses  the  Primary 
Focal  Plane  by  the  tangent  of  the  slope-angle  of  the  corresponding  re- 
fracted ray. 

In  the  special  case  where  the  optical  system  consists  of  a  single 
spherical  refracting  surface,  the  Focal  Lengths  may  also  be  defined  as 
follows  : 

The  Focal  Lengths  of  a  Spherical  Refracting  Surface  are  equal  to  the 
abscisses  of  the  vertex  A  with  respect  to  each  of  the  Focal  Points;  that  is, 
/  =  FA,  er  =  E'A,  as  above  stated. 

The  Focal  Lengths  /  and  e'  of  a  spherical  refracting  surface  may 
easily  be  expressed  in  terms  of  the  radius  r  =  A  C.  Thus,  in  Fig.  60, 
from  the  two  pairs  of  similar  triangles  AFYf,  G'Y^G  and  E'AYe, 
G'  YfYe  we  obtain  the  following  proportions  : 

FA  :  YfGf  =  AYf:  G'G,    E'A  :  G'Y,  =  A  Ye  :  7,7.; 
and  since 

YG'  =  AC  =  r,     AY  =  G'G  =  CG  -  CG',     CG  :  CG'  =  »':», 

we  derive  immediately  the  following  formulae  for  the  magnitudes  of 
the  Focal  Lengths  in  terms  of  the  radius  r: 

f  =  ~—  r,    e'=-  -?—  r;  (71) 

J      n'  —  n  n'  —  n 


158  Geometrical  Optics,  Chapter  V.  [  §  125. 

whence  also  we  find : 

n'f+ne'  =  o,  (72) 

which  is  equivalent  to  formula  (69) ;  and  also : 

f+e'  +  r  =  o-,  (73) 

which  is  equivalent  to  formula  (68). 

ART.  37.     THE  IMAGE-EQUATIONS  IN  THE  CASE  OF  THE  REFRACTION  OF 
PARAXIAL  RAYS  AT  A  SPHERICAL  SURFACE. 

125.  The  Abscissa-Equation  in  Terms  of  the  Constants  n,  n'  and  r. 
If  the  vertex  A  of  the  spherical  refracting  surface  is  taken  as  origin 
of  a  system  of  rectangular  axes  whose  #-axis  is  the  optical  axis  deter- 
mined by  the  centre  C  and  the  vertex  A,  the  co-ordinates  of  an  Object- 
Point  Q  may  be  denoted  by  u,  y  and  of  the  corresponding  Image-Point 
G'by*',/;  thus: 

AM  =  u,     AM'  =  u',     MQ  =  y,     M'Q'  =  /. 

The  problem  is  to  determine  u',  yr  in  terms  of  u,  y. 
Since 

CM  =  CA  +  AM  =  u-r,     CM'  =  CA  +  AM'  =  u'  -  r , 
equation  (66)  may  be  written  in  the  following  form: 

u  —  r     u       n' . 
u'  —  r'  u'      n' 
or,  finally: 

n'      n      n'  —  n 

J'-u  =  —-  (74) 

To  every  value  of  u  comprised  between  u  =  —  oo  and  u  =  +  oo, 
we  obtain  by  this  equation  a  corresponding  value  of  the  abscissa  u' \ 
thus,  to  every  axial  Object- Point  M  there  corresponds  one,  and  only  one, 
axial  Image- Point  M'.  This  linear  equation  connecting  the  abscissae  of 
conjugate  axial  points  in  the  case  of  the  refraction  of  paraxial  rays  at 
a  spherical  surface  is  one  of  the  most  important  formulae  of  Geometri- 
cal Optics.  It  is  entirely  independent  of  the  special  law  of  refraction 
known  as  SNELL'S  Law;  for  if  the  angles  of  incidence  and  refraction 
a,  a'  are  connected  by  any  equation  of  the  form  f(a,  a')  =  o,  wherein 
it  is  assumed  that  the  angles  denoted  by  a,  a'  are  small,  it  is  easy  to 
show  that  the  limiting  value  of  the  ratio  a/ a'  will  be  a  constant  which 
may  be  denoted  by  n' jn\  in  which  case  we  shall  derive  formula  (74) 


§  126.]  Reflexion  and  Refraction  of  Paraxial  Rays.  159 

as  the  most  general  expression  of  the  relation  between  conjugate  points 
of  any  paraxial  ray  which  passes  through  the  centre  C  of  the  spherical 
surface.  In  a  supplement  to  this  chapter  it  will  be  shown  that  this 
equation  is  the  analytical  expression  of  Central  Collineation  in  a  Plane. 
126.  The  so-called  Zero-Invariant.  If  according  to  the  convenient 
method  of  notation,  introduced  by  ABBE,  we  denote  the  difference  of 
the  values  of  an  expression  before  and  after  refraction  at  a  spherical 
surface  by  the  symbol  A  written  before  the  expression,  formula  (74) 
may  be  written  also  in  the  following  abbreviated  form  : 


A-  =  -An.  (75) 

u       r 


The  magnitude 


which  has  the  same  value  before  and  after  refraction  at  the  spherical 
surface,  is  called  the  "Zero-Invariant"  or  the  invariant  in  the  case  of 
the  refraction  of  paraxial  rays  at  a  spherical  surface.  This  magnitude 
denoted  here  by  /  plays  an  important  part  in  the  Theory  of  Spherical 
Aberrations,  and  the  following  formulae,  all  easily  derived  from  (76), 
will  be  found  useful  in  the  investigations  of  that  theory.  For  example, 
we  obtain: 

A-  =  -7A-;  (77) 

u  n 

and,  also: 

nu      r     n  n2 

Again,  we  find: 

Combining  formulae  (77)  and  (79),  we  obtain: 


and  combining  formulae  (78)  and  (79) : 

/A—  =  J-A--A\-  (81) 

nu       r     n          u 

Moreover,  if  6,  0'  denote  the  slopes  of  a  ray  before  and  after  re- 
fraction at  a  spherical  surface,  and  if  a,  a'  denote  the  angles  of  incidence 


160  Geometrical  Optics,  Chapter  V.  [  §  127. 

and  refraction,  and,  finally,  if  <p  denotes  the  central  angle  (<p  =  Z.BCA, 
Fig.  51),  then,  as  in  formula  (60): 

a  —  B  =  OL   —  6'  =  tp. 

In  the  case  of  Paraxial  Rays  where  these  angles  are  all  so  small  that 
we  may  neglect  powers  above  the  first,  we  have  (see  §  108): 

* 


where  h  =  DB  (Figs.  58  and  59)  denotes  the  incidence-height  of  the 
ray.  From  these  relations  we  obtain  easily : 

«-^.    ct-%,  (83) 

n  '  n1  w; 

127.  The  Lateral  Magnification.  The  ratio  Y  =  y'/y  is  called  the 
Lateral  Magnification  of  the  spherical  refracting  surface  with  respect 
to  the  axial  Object- Point  M.  Referring  to  Figs.  62  and  63,  we  see 
that  we  have  the  proportion: 

M'Q'  :  MQ  =  CM'  :  CM, 
and,  consequently: 

yf      u'  -r 
y        u  —  r 

This  equation,  together  with  formula  (74),  enables  us  to  write  the 
transformation-formulae  between  Object-Space  and  Image-Space  as 
follows : 

,  n'ru  ,  nry 

whereby,  being  given  the  co-ordinates  u,  y  of  the  Object-Point  Q,  we 
can  find  the  co-ordinates  u' ,  y'  of  the  corresponding  Image-Point  Q'. 
The  formula  for  the  Lateral  Magnification  Y  may  also  be  written 
as  follows: 

Y=y^?u''  ^ 

whence  we  see  that  the  Lateral  Magnification  Y  is  a  function  of  the 
abscissa  u,  and  that  it  is  independent  of  the  absolute  magnitude  of 
the  ordinate  y.  For  a  given  pair  of  conjugate  planes  at  right  angles 
to  the  axis  of  a  spherical  refracting  surface,  the  ratio  denoted  by  Y 
is  constant,  but  it  is  different  for  different  pairs  of  conjugate  planes. 


§  129.]  Reflexion  and  Refraction  of  Paraxial  Rays.  161 

128.  The  Image-Equations  in  Terms  of  the  Focal  Lengths  /,  e'  . 
If>  by  means  of  formulae  (72)  and  (73),  we  eliminate  n,  n'  and  r  from 
the  formulae  (84),  the  Image-Equations  for  the  Refraction  of  Paraxial 
Rays  at  a  Spherical  Surface  may  be  obtained  also  in  the  following 
forms  : 

-  +  4=  -I,     -=7T—;  (86) 

u      u'  y       f  +  u 

wherein  the  constants  which  determine  the  spherical  refracting  surface 
are  the  two  focal  lengths/  and  e1  '. 

If,  instead  of  taking  the  vertex  A  as  the  origin  of  abscissae,  both  in 
the  Object-Space  and  in  the  Image-Space,  we  take  the  two  Focal 
Points  F  and  E'  as  origins  for  the  Object-Space  and  Image-Space,  re- 
spectively, we  may  put: 

FM  =  x,    E'M'  =  x'\ 

so  that  the  co-ordinates  of  the  conjugate  points  Q,  Q'  referred  to  axes 
with  origins  at  F,  E'  will  be  #,  y  and  #',  y',  respectively.     Evidently, 

u  =  AM  =  AF+FM  =  x-f,     u'  =  AM'  =  AE'  +  E'M'  =  x'  -  e'; 

and  substituting  these  values  in  place  of  u  and  u'  in  equations  (86), 
we  obtain  the  Image-Equations  in  their  simplest  forms,  as  follows: 


129.  The  case  of  the  Reflexion  of  Paraxial  Rays  at  a  Spherical 
Mirror,  which  was  treated  at  length  in  Arts.  33  and  34,  may  be  re- 
garded as  a  special  case  of  the  Refraction  of  Paraxial  Rays  at  a  Spheri- 
cal Surface.  Thus,  according  to  the  general  principle  explained  in 
§  26,  we  have  merely  to  put  nf  =  —  n  in  the  formulae  of  Arts.  35-37 
in  order  to  derive  at  once  the  corresponding  formulae  of  Reflexion. 
Thus,  for  example,  if  we  put  nf  =  —  n  in  formulae  (71),  we  obtain/  =  e' 
=  r/2;  which  shows  that  the  two  Focal  Points  F,  E'  coincide  in  the 
case  of  a  spherical  mirror  (§  112). 

Another  interesting  special  case  that  may  be  remarked  here  also 
is  obtained  by  putting  r  =  oo  ;  in  which  case  we  shall  obtain  the 
formulae  for  the  Refraction  of  Paraxial  Rays  at  a  Plane  Surface;  thus 
we  find: 


which  will  be  recognized  as  the  same  as  the  results  obtained  in  §  53. 
12 


162  Geometrical  Optics,  Chapter  V.  [  §  130. 

The  last  of  these  equations  is  of  special  interest,  for  it  shows  that  the 
Focal  Points  F  and  E'  of  a  refracting  plane  are  themselves  the  infi- 
nitely distant  points  of  the  two  ranges  of  conjugate  axial  points.  Hence, 
to  a  bundle  of  parallel  incident  rays  refracted  at  a  plane  there  corre- 
sponds a  bundle  of  parallel  refracted  rays.  Any  optical  system  which 
treats  parallel  incident  rays  in  this  way  is  called  a  Telescopic  System 
— a  name  which  is  derived  from  the  fact  that  the  Focal  Points  of  a 
telescope  are  both  at  infinity. 

III.    SUPPLEMENT:   CONTAINING  CERTAIN  SIMPLE  APPLICATIONS  OF 
THE  METHODS  OF  PROJECTIVE  GEOMETRY. 

ART.  38.   CENTRAL  COLLINEATION  OF  TWO  PLANE-FIELDS. 

130.  In  the  investigation  of  the  refraction  (or  reflexion)  of  paraxial 
rays  at  a  spherical  surface,  we  have  seen  that  the  imagery  is  ideal; 
so  long  at  least  as  the  rays  of  light  are  supposed  to  be  monochromatic, 
so  that  the  refractive  indices  n,  n'  have  fixed  values.  Thus,  to  a  homo- 
centric  bundle  of  Object-Rays  there  corresponds  always  a  homocentric 
bundle  of  Image-Rays,  and  to  each  point  of  the  Object-Space,  within 
the  region  of  the  paraxial  rays,  there  corresponds  one,  and  only  one, 
point  of  the  Image-Space.  This  unique  point-to-point  correspon- 
dence by  means  of  rectilinear  rays  between  the  Object-Space  and  the 
Image-Space,  which  is  the  fundamental  and  essential  requirement 
of  Optical  Imagery,  is  called  in  the  modern  geometry  "CoMneation". 
Thus, 

Two  space-systems  S  and  S'  are  said  to  be  "collinear"  with  each  other, 
if  to  every  point  P  of  S  there  corresponds  one,  and  only  one,  point  P'  of 
S',  and  to  every  straight  line  of  2  which  goes  through  P  there  corresponds 
one  straight  line  of  S'  which  goes  through  P'. 

In  the  theory  of  optics  these  two  spaces  S  and  S'  are  designated 
as  the  Object-Space  and  the  Image-Space.  They  are  not  to  be  thought 
of  as  separate  and  distinct  parts  of  space;  they  penetrate  and  include 
one  another,  so  that  a  point  or  a  ray  may  be  regarded  as  belonging  to 
either  of  the  two  space-systems. 

Inasmuch  as  the  problem  of  the  refraction  of  paraxial  rays  affords 
a  simple  and  at  the  same  time  a  very  useful  application  of  the  elegant 
methods  of  the  modern  geometry,  it  is  proposed  to  give  here  a  special 
investigation  of  it  from  this  point  of  view;  especially,  too,  because 
this  study  will  prove  a  good  introduction  to  the  general  theory  of 
Optical  Imagery  which  is  treated  at  length  in  Chapter  VII. 

Since  the  optical  axis  of  the  spherical  surface  is  an  axis  of  symmetry 


§  131.]  Reflexion  and  Refraction  of  Paraxial  Rays.  163 

for  both  the  Object-Space  and  the  Image-Space,  it  will  suffice,  as  we 
have  seen,  to  investigate  the  imagery  in  any  meridian  plane  of  the 
spherical  surface;  that  is,  in  any  plane  containing  the  optical  axis. 
In  this  plane  in  space  we  have  two  collinear  plane-fields,  one  belonging 
to  the  Object-Space  and  one  belonging  to  the  Image-Space,  which 
correspond  with  each  other  point  by  point  and  ray  by  ray.  The  total- 
ity of  all  the  points  and  straight  lines  situated  in  an  infinitely  extended 
plane  is  what  is  here  meant  by  the  term  "plane-field". 

The  distinguishing  characteristics  of  the  kind  of  Collineation  which 
we  have  in  the  case  of  the  Refraction  of  Paraxial  Rays  at  a  Spherical 
Surface  may  be  said  to  be  two  in  number,  although,  indeed,  one  is  a 
consequence  of  the  other.  These  characteristics  are  contained  in  the 
following  statements: 

(1)  If  Q,  Q'  are  a  pair  of  corresponding,  or  conjugate,  points  the 
straight  line  QQ'  passes  through  the  centre  C  of  the  spherical  refracting 
surface;  or,  the  straight  lines  joining  pairs  of  conjugate  points  all  inter- 
sect in  one  point  ( C) . 

(2)  Since  an  incident  ray  and  its  corresponding  refracted  ray  meet 
in  the  spherical  refracting  surface,  and,  moreover,  since  we  are  con- 
cerned here  only  with  paraxial  rays,  which,  therefore,  meet  the  spheri- 
cal surface  at  points  infinitely  near  to  its  vertex  A ,  so  that  the  straight 
line  (y)  in  the  meridian  plane  which  is  tangent  to  the  spherical  surface 
at  A  may  be  regarded  as  the  section  of  the  surface  made  by  this  plane 
(see  §  113);  it  follows,  therefore,  that  any  pair  of  corresponding  rays 
of  the  two  collinear  plane-fields  will  meet  in  this  straight  line  (y) . 

When  two  collinear  plane-fields  are  so  situated  relative  to  each  other 
that  they  have  in  common  a  self-corresponding  range  of  points,  we  have 
the  special  case  of  the  "Central  Collineation"  of  two  plane-fields. 
The  straight  line  (y)  which  corresponds  with  itself  point  by  point  is 
called  the  "Axis  of  Collineation".  The  point  C  through  which  every 
straight  line  joining  a  pair  of  corresponding  points  passes  is  called  the 
"Centre  of  Collineation".  This  point  C  is  a  "double  point"  or  self- 
corresponding  point  of  the  two  collinear  plane-fields.  Hence,  every 
straight  line  drawn  through  C  contains  two  double  points,  viz.,  the  Centre 
of  Collineation  itself  and  the  point  where  the  straight  line  intersects 
the  Axis  of  Collineation. 

131.  Projective  Relation  of  Two  Collinear  Plane-Fields.  If  P,  Q, 
R,  S  (Fig.  66)  are  a  range  of  four  points  lying  on  a  straight  line  s  of 
one  of  the  plane-fields,  the  points  P',  Q',  R',  S'  conjugate  to  P,  Q,  R,  S, 
respectively,  will  be  ranged  along  the  corresponding  straight  line  s' 
of  the  collinear  plane-field,  and  it  is  easy  to  show  that  we  have  the 


164  Geometrical  Optics,  Chapter  V. 

following  relation : 

(PQRS)  =  (P'Q'R'S'); 


[§131. 


FIG.  66. 

CENTRAL  COLLINEATION  OP  Two  PLANE- 
FIELDS.  The  centre  of  collineation  (C)  and  the 
axis  of  collineation  ( y)  are  the  centre  and  axis  of 
perspective;  so  that  if  s.  /  are  a  pair  of  corre- 
sponding rays, 

(PQRS)  =  (P'Q'R'S'). 


that  is,  two  collinear  plane-fields  are  in  "projective"  relation  to  each  other. 
The  proof  of  this  is  especially  simple  when  we  have  Central  Collinea- 
tion of  the  two  plane-fields;  for 
then  every  straight  line  joining 
a  pair  of  corresponding  points 
passes  through  the  centre  of 
collineation  C,  and  hence  the 
two  plane-fields  in  this  case 
will  be  in  perspective;  whence 
it  follows  that  any  two  corre- 
sponding straight  lines  s,  s'  will 
intersect  a  pencil  of  rays  with 
its  vertex  at  C  in  two  projective 
ranges  of  points. 

In  case,  however,  the  ray  5  itself  passes  through  the  centre  C,  so 
that  5,  s'  are,  therefore,  a  pair  of  self-corresponding  rays  (Fig.  67),  the 
above  proof  of  the  projective  relation  of  s,  sf  will  not  be  applicable. 
In  such  a  case  we  may  proceed  as  follows : 

Through  the  point  C  draw  any  other  straight  line,  and  take  on  it 
a  point  O.  Connect  0  by  straight  lines  with  the  Object-Points  P,  Q, 
R,  S  ranged  along  the  straight  line  5.  The  straight  lines  joining  the 
corresponding  Image-Points  P',  Qf,  Rf,  S'  ranged  along  the  straight 
line  s'  with  the  points  where  the  straight  lines  PO,  QO,  RO,  5O,  respec- 
tively, intersect  the  axis  of  collineation  (y)  will  all  pass  through  the 
point  0'  conjugate  to  0,  which  is  a  point  of  the  straight  line  joining 
0  and  C.  Since,  therefore,  the  Object- Ray  OC  and  the  corresponding 
Image-Ray  0'  C  coincide  in  the  straight  line  joining  0  and  O',  the  pen- 
cils of  rays  0,  0'  are  in  perspective  with  each  other ;  so  that  for  con- 
jugate points  P,  Q,  R,  S  and  P',  Q',  R',  Sf  of  a  central  or  self-correspond- 
ing ray  s  (or  s')  we  have  also  the  projective  relation,  characterized  by  the 
equation : 

(PQRS)  =  (P'Q'R'S'). 

The  self-corresponding  ray  at  right  angles  to  the  axis  of  collineation 
(y)  coincides  with  the  optical  axis  of  the  system.  This  ray  will  be 
designated  as  the  ray  x  of  the  Object-Space  and  the  ray  x'  of  the 
Image-Space.  And  the  point  A  where  it  crosses  the  axis  of  collinea- 
tion will  be  selected,  in  the  special  case  of  Central  Collineation,  as  the 


§  132.] 


Reflexion  and  Refraction  of  Paraxial  Rays. 


165 


most  convenient  point  for  the  origin  of  a  system  of  rectangular  co-ordi- 
nates, the  axes  whereof  are  the  optical  axis  and  the  axis  of  collineation. 


FIG.  67. 

CENTRAL  COLLINEATION  OF  Two  PLANE- FIELDS.  Construction  of  Conjugate  Points  of  a  self- 
corresponding  or  central  ray  s  (or  /).  The  centre  of  collineation  is  at  C;  the  axis  of  collineation, 
or  the  perspective-axis,  is  the  straight  line  designated  by  y.  O,  Cf  are  a  pair  of  conjugate  points  on 
any  straight  line  passing  through  C.  The  points  P.  Q,  R,  Sof  the  range  of  Object- Points  s  are  pro- 
jected by  a  pencil  of  rays  from  O,  and  the  conjugate  points  P' ,  Q' ,  K! ',  Sf  of  the  range  of  image- 
points  /  are  projected  by  a  pencil  of  rays  from  Of  which  is  in  perspective  with  the  pencil  O.  The 
points  /',  J  are  points  of  s',  s  conjugate  to  the  infinitely  distant  points  /,  J'  of  s,  sf,  respectively. 
The  self-corresponding  ray  x  (or  x')  which  meets  the  axis  of  collineation  at  A  at  right  angles  is 
the  optical  axis  ;  and  the  straight  lines  parallel  to  the  axis  of  collineation  which  are  drawn  through 
/and  /'  and  which  meet  the  optical  axis  at  right  angles  at  P  and  E' ,  respectively,  are  the  two  focal 
lines. 

132.     Geometrical  Constructions. 

(i)  Given  the  axis  of  collineation  (y)  and  the  centre  of  collineation  (C), 
together  with  the  positions  of  two  conjugate  points  P,  Pf:  it  is  required  to 
construct  the  Image- Point  Qf  of  a  given  Object- Point  Q. 

Through  the  two  given  Object-Points  P,  Q  draw  the  straight  line  s 
meeting  the  axis  of  collineation  in  the  double  point  B.  Suppose  (i) 
that  the  straight  line  s  joining  P,  Q  does  not  pass  through  the  centre  C, 
as  in  Fig.  66.  This  is  the  general  case.  The  straight  line  sf  corre- 
sponding to  s  will  connect  B  with  the  given  point  P',  and  this  straight 
line  must  also  pass  through  the  point  Q'  conjugate  to  Q.  But  Qr 
must  also  lie  on  the  self-corresponding  ray  which  goes  through  Q  and 
the  centre  C;  and  hence  the  Image-Point  Q'  will  be  uniquely  deter- 
mined by  the  intersection  of  the  straight  lines  BP'  and  QC.  Again 
suppose  (ii)  that  the  straight  line  s  joining  P,  Q  passes  through  the 
centre  C,  as  in  Fig.  67;  so  that  5  (or  5')  is  a  self-corresponding  ray. 


166  Geometrical  Optics,  Chapter  V.  [  §  132. 

This  is  a  special  case  of  great  importance.  In  this  case  the  above  con- 
struction fails,  and  we  may  proceed,  therefore,  as  follows:  From  P 
and  C  draw  two  straight  lines  intersecting  in  a  point  O.  Join  by  a 
straight  line  the  point  where  PO  meets  the  axis  of  collineation  with  the 
given  point  Pr  conjugate  to  P,  and  let  O'  designate  the  point  where  this 
straight  line  meets  the  straight  line  CO.  Join  QO  by  a  straight  line 
and  from  the  point  where  QO  meets  the  axis  of  collineation  draw 
through  0'  a  straight  line,  which  will  meet  s'  in  the  Image-Point  Q' 
conjugate  to  the  given  Object-Point  Q. 

(2)  Construction  of  the  so-called  "Flucht"  Points  J  and  I'  of  any 
central,  or  self-corresponding,  ray  s  (or  s') ;  being  given,  as  before,  the 
axis  of  collineation  (y),  the  centre  of  collineation  (C)  and  the  pair  of 
conjugate  points  P,  Pf . 

The  Image-Point  /'  conjugate  to  the  infinitely  distant  Object- Point 
I  of  the  pencil  of  parallel  object-rays  of  which  the  self-corresponding 
ray  is  the  central  ray  s  (Fig.  67)  will  be  a  point  on  s'  which  may  be 
constructed  exactly  as  was  explained  above.  For  example,  knowing 
the  positions  of  P,  Pf ,  we  can  locate  the  positions  of  a  pair  of  conjugate 
points  0,  0' ,  as  was  done  above.  A  straight  line  drawn  through  0 
parallel  to  5  will  go*  through  the  infinitely  distant  point  J  of  s.  The 
straight  line  joining  the  point  where  01  meets  the  axis  of  collineation 
with  the  point  0'  will  intersect  s'  in  the  Image-Point  /'  conjugate  to 
the  infinitely  distant  Object-Point  J.  German  writers  on  geometry 
call  this  point  /'  the  "Flucht"  Point  of  the  ray  s'. 

Similarly,  the  "Flucht"  Point  /  of  the  Object-Ray  5  is  that  point 
of  this  range  which  corresponds  with  the  infinitely  distant  point  J'  of 
the  Image-Ray  sf.  It  may  be  constructed  in  a  way  precisely  analogous 
to  the  construction  of  /'  above,  in  the  manner  indicated  in  the  diagram. 

The  "Flucht"  Points  J  and  I'  are,  in  general,  actual,  or  finite,  points 
of  the  projective  ranges  of  points  5  and  s',  respectively.  In  particular, 
the  "Flucht"  Points,  designated  by  F  and  E',  of  the  self-corresponding 
ray  x,  x' ,  which  coincides  with  the  optical  axis,  are  the  points  called 
the  Focal  Points  of  the  optical  system  (§  120). 

(3)  Given  the  axis  of  collineation  (y),  together  with  the  positions  of  the 
"Flucht"  Points,  J  and  I' ,  of 'any  central  ray  s,  s' ,  to  construct  the  Image- 
Point  P'  corresponding  to  a  given  Object- Point  P  of  s. 

Take  any  point  0  (Fig.  67),  and  through  0  draw  the  straight  line 
01  parallel  to  s;  and  draw  the  straight  line  joining  with  /'  the  point 
where  OI  meets  the  axis  of  collineation.  Draw  the  straight  lines  JO, 
PO,  and  from  the  point  where  JO  meets  the  axis  of  collineation  draw  a 
straight  line  parallel  to  s'  meeting  in  0'  the  straight  line  drawn  through 


§  132.] 


Reflexion  and  Refraction  of  Paraxial  Rays. 


167 


/'.  The  straight  line  which  joins  with  0'  the  point  where  PO  meets 
the  axis  of  collineation  will  meet  sr  in  the  Image-Point  P'  conjugate 
to  the  Object-Point  P. 

In  particular,  knowing  the  positions  of  the  two  Focal  Points  F 
and  E'  on  the  optical  axis,  and  knowing  also  the  position  of  the  axis 
of  collineation,  we  may,  as  above,  construct  any  pair  of  conjugate  axial 
points  M,  M'. 

(4)  Given  the  axis  of  collineation  (y)  and  the  centre  of  collineation  (C), 
together  with  the  positions  of  two  conjugate  points  P,  P'  it  is  required  to 
construct  the  image-ray  vf  corresponding  to  a  given  object-ray  v. 

Let  H  (Fig.  68)  designate  the  double  point  where  the  given  object- 
ray  meets  the  axis  of  collineation.  Through  the  given  Object-Point 


FIG.  68. 

CENTRAL  COLLINEATION  OF  Two  PLANE-FIELDS.    Construction  of  Image-Ray  i/  conjugate  to 
given  Object-Ray  v ;  also,  construction  of  the  "  Flucht "  I^ines  or  Focal  I^inesy,  /. 

P  draw  any  ray  s  meeting  the  given  ray  v  in  a  point  Q  and  the  axis  of 
collineation  in  a  point  B.  The  point  of  intersection  of  QC  and  BPr 
will  determine  the  Image-Point  Q'  conjugate  to  the  Object- Point  Q't 
and  hence  the  straight  line  HQf  will  be  the  image-ray  vf  conjugate  to 
the  given  object-ray  v. 

(5)  //  the  given  object-ray  in  (4)  is  the  infinitely  distant  straight  line  e 
of  the  Object- Plane,  we  can  construct  the  conjugate  straight  line  e'  oj  the 
Image- Plane,  as  follows: 

The  point  of  intersection  of  the  infinitely  distant  straight  line  e  of 
the  Object-Plane  with  the  axis  of  collineation  (y)  is  the  infinitely  dis- 
tant point  T  (Fig.  68)  of  y;  and  hence  e'  will  be  parallel  to  y.  Any 
ray  5  drawn  through  the  given  Object-Point  P  will  meet  the  infinitely 
distant  straight  line  e  of  the  Object-Plane  in  the  infinitely  distant 
point  5  of  s.  If  the  object-ray  s  meets  y  in  B,  the  corresponding 


168  Geometrical  Optics,  Chapter  V.  [  §  133. 

image-ray  s'  will  be  the  straight  line  BPf,  and  a  straight  line  drawn 
through  the  centre  C  parallel  to  5  will  determine  by  its  intersection 
with  s'  the  Image-Point  Sr  conjugate  to  the  infinitely  distant  Object- 
Point  5.  The  straight  line  drawn  through  Sf  parallel  to  y  will,  there- 
fore, be  the  image-ray  e'  conjugate  to  the  infinitely  distant  object-ray  e. 

This  straight  line  e'  which  is  conjugate  to  the  infinitely  distant 
straight  line  e  of  the  Object- Plane  is  called  in  Optics  the  Focal  Line  of 
the  Plane-Field  of  the  Image-Space  (see  §123).  Since  e'  passes  through 
the  point  S',  which  is  the  "Flucht"  Point  of  any  ray  of  the  plane-field 
of  the  Image-Space,  it  follows  that  the  Focal  Line  e'  is  the  locus  of  the 
"Flucht"  Points  of  all  the  image-rays  in  this  plane-field. 

In  a  precisely  similar  way,  we  can  construct  also  the  straight  line  f 
in  the  plane-field  of  the  Object-Space  which  is  conjugate  to  the  infinitely 
distant  straight  line  f  of  the  plane-field  of  the  Image-Space,  and  which 
may,  likewise,  be  defined  as  the  locus  of  the  "Flucht"  Points  of  all  the 
rays  in  the  plane-field  of  the  Object-Space. 

The  focal  lines  /,  e'  are,  in  general,  actual,  or  finite,  straight  lines. 
They  are  both  parallel  to  the  axis  of  collineation,  and  perpendicular, 
therefore,  to  the  optical  axis. 

133.  The  Invariant  in  the  Case  of  Central  Collineation.  Since 
all  the  rays  of  the  pencil  C  are  self-corresponding,  each  of  these  rays 
is  the  base  of  two  projective  ranges  of  points,  a  range  of  Object-Points 
and  a  range  of  corresponding  Image-Points.  Moreover,  to  each  of 
these  self -corresponding  rays  belongs  a  pair  of  double,  or  self -corresponding, 
points  (§  130) ;  one  of  these  double  points  being  the  centre  of  collinea- 
tion itself  and  the  other  the  point  where  the  ray  crosses  the  axis  of  col- 
lineation. 

Similarly,  each  point  on  the  axis  of  collineation  is  the  common 
vertex  of  two  projective  pencils  of  rays,  viz.,  a  pencil  of  object-rays 
and  a  pencil  of  corresponding  image-rays ;  and  each  pair  of  such  pencils 
of  corresponding  rays  contains  two  self -corresponding  rays,  of  which  the 
axis  of  collineation  itself  is  one,  and  the  ray  joining  the  common  ver- 
tex of  the  two  pencils  with  the  centre  of  collineation  is  the  other. 

Let  P,  Pf  (Fig.  69)  and  Q,  Qf  be  two  pairs  of  conjugate  points  of  the 
self-corresponding  ray  s,  s',  and  let  U  designate  the  double  point  where 
this  ray  crosses  the  axis  of  collineation  (y).  Since  the  ray  s,  s'  is  the 
common  base  of  two  projective  ranges  of  points,  the  double  ratio  of 
the  four  Object-Points  C,  U,  P,  Q  on  s  is  equal  to  the  double  ratio 
of  the  four  corresponding  Image-Points  C,  U,  P',  Q'  on  s' ';  that  is, 

(CUPQ)  =  (CUP'Q')\ 


§  133.] 


Reflexion  and  Refraction  of  Paraxial  Rays. 


169 


whence  it  follows  immediately  that  we  have  also: 

(CUPP')  =  (CUQQ')\ 

and,  consequently,  the  double  ratio  of  any  pair  of  conjugate  points  P,  P' 
with  the  two  self -cor  responding  points  C,  U  of  the  two  protective  ranges 
of  points  which  have  the  common  base  PPf  has  a  constant  value,  which 
is  independent  of  the  positions  of  the  conjugate  points  P,  P'. 

Let  M,  M'  be  any  other  pair  of  conjugate  points  not  situated  on 
the  straight  line  PP';  for  example,  it  will  be  perfectly  general  if  we 


FIG.  69. 

CENTRAL  COLLINEATION  OF  Two  PLANE-FIELDS. 
(CUPP')  =  (CUQQ1)  =  (CAMM')  =  (CALL')  =  (CAEE')  =  (CAFf). 

take  the  points  M,  M'  on  the  optical  axis  x,  x'  which  crosses  the  axis 
of  collineation  (;y)  at  the  point  A.  Let  the  two  corresponding  rays 
MP,  M' P'  intersect  in  a  point  H  on  the  axis  of  collineation.  It  is 
obvious  immediately  that  the  two  ranges  of  points  C,  U,  P,  P'  and 
C,  A,  M,  M'  are  in  perspective,  since  they  are  both  sections  of  the 
pencil  of  rays  which  has  its  vertex  at  H.  Hence,  the  double  ratios 
of  each  of  these  ranges  of  four  points  are  equal ;  and  if  we  denote  the 
value  of  this  double  ratio  by  the  symbol  c,  we  have  the  following  re- 
markable relations: 


c  =  (CUPPr)     =  (CUQQ'}  =  etc., 
=  (CAMM')  =(CALL')  =  etc., 
=  (CAFF')     =  CF  :  AF, 
=  (CAEE')    =  AE'  :  CE'; 


(88) 


170  Geometrical  Optics,  Chapter  V.  [  §  134. 

where,  as  heretofore,  F  and  Ef  designate  the  positions  of  the  two  Focal 
Points,  and  F'  and  E  designate  the  infinitely  distant  points  of  x'  and 
x,  respectively. 

The  most  striking  characteristic  of  the  Central  Collineation  of  two 
plane'-fields  consists,  therefore,  in  the  fact  which  we  have  here  dis- 
covered, that  it  has  an  invariant: 

The  Double  Ratio  of  any  pair  of  conjugate  points  of  a  self  -corresponding 
ray  and  the  two  double  points  of  the  ray  has  the  same  value  for  all  such 
rays. 

The  value  of  this  invariant,  as  above  stated,  is: 

CF  _  AE'  % 
C~  AF~  CE'' 
accordingly, 

CA+AF     AC+CE'  AE' 


AF  CE'         ~  CA  +  AE'  ~    ' 

which  gives: 

FA  =  CE',    E'A  =  CF,     ff  =  -  c.  (89) 

These  relations  are  likewise  characteristic  of  Central  Collineation. 
The  first  two  of  formulse  (89)  —  which  may  be  derived  also  directly 
from  the  equation  (  CFA  E)  =  (  CF'  A  E')  —  are  identical  with  formulae 
(67)  which  were  obtained  for  the  special  case  of  the  Refraction  of 
Paraxial  Rays  at  a  Spherical  Surface;  whereas  the  third  equation  cor- 
responds with  the  relation  given  in  formula  (69). 

134.  The  Characteristic  Equation  of  Central  Collineation.  In  par- 
ticular, if  M,  M'  designate  the  positions  of  any  two  conjugate  points 
of  the  optical  axis,  the  relation 

(CAMM')  =  c 


may  be  written  in  the  following  form  : 

c       i      c  —  i 


,     N 

(9o) 


where  the  symbols  u,  u'  and  r  denote  the  abscissae,  with  respect  to  the 
point  A  as  origin,  of  the  .points  M,  M'  and   C,  respectively;  thus, 

u  =  AM,     u'  =  AM',     r  =  AC. 

This  equation,  which  expresses  for  the  case  of  any  Central  Collineation 
the  relation  between  the  abscissae  of  conjugate  axial  points,  is  a  per- 


§  134.]  Reflexion  and  Refraction  of  Paraxial  Rays  171 

fectly  general  expression  of  the  one-to-one  correspondence  of  two  pro- 
jective  ranges  of  points  lying  upon  the  same  straight  line.  The  cases 
which  occur  in  Optics  are  comparatively  restricted;  we  shall  proceed 
to  examine  them. 

//  the  sign  of  the  invariant  (c)  is  positive,  the  conjugate  points  M,  M' 
are  not  "separated",  in  the  geometrical  sense,  by  the  axis  of  col- 
lineation  (y)  and  the  centre  of  collineation  (C).  That  is,  for  c  >  o, 
the  points  M  and  M'  are  either  both  situated  between  C  and  A,  or 
neither  of  them  lies  between  C  and  A.  In  other  words,  the  points 
C,A,M,  M'  are  what  is  called  a  "hyperbolic  throw",  (CAMM')  >  o. 
This  case  occurs  always  when  the  rays  are  refracted  from  one  medium 
to  another ;  so  that  in  Optics  a  positive  value  of  c  indicates  Refraction; 
whereas,  on  the  contrary,  whenever  the  light-rays  are  reflected  at  a  mirror, 
the  imagery  is  of  a  kind  that  corresponds  to  a  negative  value  of  c  (c  <  o) ; 
in  which  case  one  of  the  points  M  or  M'  will  lie  between  C  and  A ,  but 
not  the  other  point.  In  this  latter  case  the  points  C,  A,  M,  Mr  are  an 
"elliptical  throw",  (CAMM')  <  o. 

Case  I.     Refraction  of  Paraxial  Rays;  c  >  o. 

A.  Suppose,  first,  that  r  =  AC  is  not  equal  to  zero;  that  is,  that 
the  centre  of  collineation  (C)  does  not  lie  on  the  axis  of  colline- 
ation (y). 

This  is  the  case  of  the  Refraction  of  Paraxial  Rays  at  a  Spherical 
Surface,  which  has  been  specially  treated  in  this  chapter.  The  in- 
variant c  in  formula  (90)  is  identical  in  value  with  the  relative  index 
of  refraction  (n'/ri)  from  the  first  medium  to  the  second  medium, 
while  the  other  constant  r  denotes  here  the  radius  of  the  spherical  sur- 
face, as  will  be  seen  by  comparing  formula  (90)  with  formula  (74). 
The  points  A  and  Care,  therefore,  identical  with  the  vertex  and  centre, 
respectively,  of  the  spherical  refracting  surface. 

Several  special  cases  included  under  this  head  may  be  .briefly 
noticed : 

(1)  If  c  =  +  i  (the  value  of  r,  as  above  specified,  being  different 
from  zero),  the  relative  index  of  refraction  is  equal  to  unity  (n'  =  n). 
In  this  case  equation  (90)  gives  u'  =  u\  and,  hence,  Object-Space  and 
Image-Space  coincide  point  by  point;  in  fact,  the  two  spaces  are  identi- 
cal.    When  n'  =  n,  there  is  no  optical  difference  between  the  first 
medium  and  the  second  medium. 

(2)  The  case  when  r  =  oo .     An  infinite  value  of  r  in  this  case  means 
merely  that  the  centre  C  is  at  an  infinite  distance  away  in  the  direction 
of  a  line  at  right  angles  to  the  axis  of  collineation  (y)]  so  that  now 


172 


Geometrical  Optics,  Chapter  V. 


[  §  134. 


the  refracting  surface  is  a  plane  surface.     Formula  (90)  becomes  now 


u  =  —  u, 
n 


which  is  the  abscissa-relation  for  the  case  of  the  Refraction  of  Paraxial 
Rays  at  a  Plane  (§53  and  §  129).  Since  the  centre  of  collineation  (C) 
is  at  an  infinite  distance  in  a  direction  perpendicular  to  the  refracting 
plane,  the  trace  of  which  in  the  plane  of  the  diagram  (Fig.  70)  is  the 


FIG.  70. 

CENTRAL  COLLINEATION  OF  Two  PLANE- FIELDS  FOR  THE  CASE  WHEN  c>0  AND  r=  ».  The 
diagram  shows  the  case  when  n  =  c  >  1.  This  case  (c  >  0,  r  =  » )  is  the  case  of  the  Refraction  of 
Paraxial  Rays  at  a  Plane  Surface.  The  double  point  C  is  the  infinitely  distant  point  of  the  optical 
axis  xx* ',  and  the  two  Focal  Points  F,  E'  both  coincide  with  C. 

AM=u.    AM'  =  uf,    MQ=y  =  M'Q'  =  ye. 

axis  of  collineation  (y),  all  straight  lines  joining  pairs  of  conjugate 
points  are  parallel  to  the  abscissa-axis.  In  this  case  the  infinitely 
distant  straight  lines  of  the  two  collinear  plane-fields  must  pass  through 
the  infinitely  distant  double  point  C;  and,  therefore,  the  two  infinitely 
distant  straight  lines  must  be  a  pair  of  self -corresponding  rays,  and, 
accordingly,  the  five  points  designated  by  C,  E,  E',  F,  Ff  must  all 
be  coincident.  In  the  modern  geometry  two  collinear  plane-fields  are 
said  to  be  in  affinity  with  each  other  when  their  infinitely  distant 
straight  lines  are  conjugate  to  each  other.  Hence  in  this  case  the  two 
focal  lines  /  and  er  are  coincident  with  the  infinitely  distant  straight 
lines  e  and  /',  respectively.  In  Optics  this  type  of  imagery  is  called 
telescopic  (§  129). 

B.  A  case  of  Central  Collineation  which  is  of  much  importance  in 
Optics  is  the  case  when  the  invariant  c' =  +  i  and  the  other  constant 
r  =  o.  If  r  =  6,  the  centre  of  collineation  ( C}  is  situated  on  the  axis  of 
collineation  (y) ,  so  that  the  two  double  points  A  and  C  of  the  optical 
axis  coincide.  In  this  case  we  find : 

FA  =  AE',     or    /  +  e'  =  o; 

where  /  =  FA,  e'  =  E'A  denote  the  focal  lengths  of  the  optical  sys- 
tem. This  type  of  imagery  is  characterized,  therefore,  by  the  fact 


§  134.]  Reflexion  and  Refraction  of  Paraxial  Rays.  173 

that  the  two  focal  points  are  equidistant  from  the  axis  of  Collineation, 
and  on  opposite  sides  thereof.  In  the  following  chapter  it  will  be  seen 
that  this  is  the  imagery  obtained  by  the  refraction  of  paraxial  rays 
through  an  Infinitely  Thin  Lens,  or  through  any  number  of  such  lenses 
in  successive  contact  with  each  other. 

In  this  special  case,  the  expression  on  the  right-hand  side  of  formula 
(90)  becomes  illusory.  This  leads  us  to  remark  here  that  we  can  ob- 
tain the  abscissa-relation  of  Central  Collineation  in  another  form, 
which  is  characteristic  not  only  of  Central  Collineation,  but  of  the  col- 
linear  relation  in  general.  Thus,  since 

(MAFE)  =  (M'AF'E'), 
we  derive  the  equation: 

**'=/*', 

where  x  =  FM,  x'  =  E'M'  denote  the  abscissae  of  M,  M'  referred  to 
the  Focal  Points  F,  E',  respectively,  as  origins.  In  the  special  case 
here  under  consideration  for  which  we  have  e'  =  —  /,  this  formula 
takes  the  form: 

xx' =  -f. 

Case  II.     Reflexion  of  Paraxial  Rays;  c  <  o. 

The  only  negative  value  of  c  that  has  any  practical  significance  in 
Optics  is  the  value  c  —  —  i .  For  this  value  of  c  we  have : 

(CAMM*)  =  -  i; 

so  that  each  pair  of  conjugate  points  is  harmonically  separated  by  the 
centre  ( C}  and  the  axis  of  Collineation  (y) .  This  formula  will  be  recog- 
nized immediately  as  the  formula  for  the  Reflexion  of  Paraxial  Rays 
at  a  Spherical  Mirror  (§  in). 

Since  (CAMM')  =  -  i  =  (CAM'M),  the  two  ranges  of  points 
lying  upon  any  central  ray  are  in  "involutory  position";  so  that,  if, 
for  example,  a  point  M'  of  one  range  x'  is  conjugate  to  a  point  M  of 
the  conlocal  range  x,  the  same  point  M  regarded  now  as  a  point  of 
x'  will  be  conjugate  to  M'  regarded  as  a  point  of  x  (see  §  no).  We 
find  here: 

FA  =  E'A,     or    f  -  e'  =  o; 

so  that,  as  was  pointed  out  in  §  1 12,  the  Focal  Points  F,  E'  of  a  spherical 
mirror  are  coincident. 

Finally,  if  r  =  oo,  we  have,  for  c  =  —  i,  u'  =  —  u,  which  (see  §  50) 
is  the  formula  for  Reflexion  at  a  Plane  Mirror.^ 

1  In  connection  with  Art.  38,  see  J.  P.  C.  SOUTHALL:  The  geometrical  theory  of  optical 
imagery:  Astrophys.  Journ.,  xxiv.  (1906),  156-184. 


CHAPTER   VI. 

REFRACTION  OF  PARAXIAL  RAYS  THROUGH  A  THIN  LENS,  OR  THROUGH 
A  SYSTEM  OF  THIN  LENSES. 

ART.  39.     REFRACTION  OF  PARAXIAL  RAYS  THROUGH  A  CENTERED  SYSTEM 
OF  SPHERICAL  SURFACES. 

135.  Centered  System  of  Spherical  Surfaces.     Nearly  all  optical 
instruments  consist  of  a  combination  of  transparent  isotropic  media, 
each  separated  from  the  next  by  a  spherical  (or  plane)  surface;  the 
centres  of  these  surfaces  lying  all  on  one  and  the  same  straight  line, 
called  the  "optical  axis"  of  the  centered  system  of  spherical  surfaces, 
which  is  an  axis  of  symmetry.     The  spherical  surface  which  the  rays 
encounter  first  is  called  the  first  surface  of  the  system;  in  our  diagrams, 
in  which  the  light  is  represented  as  being  propagated  from  left  to 
right,  the  first  surface  will  be  the  one  farthest  to  the  left.     The  two 
media  separated  by  this  surface  will  be  called  the  first  and  second 
media,  respectively,  in  the  sense  in  which  the  light  travels.     If  the 
number  of  spherical  surfaces  is  m,  the  number  of  media  will  be  m  +  i> 
the  (m  -+•  i)th  medium  being  the  last  medium  into  which  the  rays 
emerge  after  refraction  (or  reflexion)  at  the  rath  surface.     The  ab- 
solute index  of  refraction  of  the  first  medium  will  be  denoted  by  nv 
(=  n'0);  and,  generally,  the  index  of  refraction  of  the  kih  medium 
(where  k  denotes  any  positive  integer  from  o  to  m)  will  be  denoted  by 
»*_!•     Thus,  the  index  of  refraction  of  the  last  medium  will  be  de- 
noted by  n'm.     The  centre  of  the  kth  spherical  surface  will  be  desig- 
nated by  Ck,  and  the  point  where  the  optical  axis  meets  this  surface, 
called  the  vertex  of -'the  surface,  will  be  designated  by  Ak.     The  cen- 
tered system  of  spherical  surfaces  is  completely  determined  provided 
we  know  the  index  of  refraction  of  each  of  the  successive  media  and 
the  positions  on  the  optical  axis  of  the  centres  and  vertices  of  the 
spherical  surfaces. 

136.  To  a  homocentric  bundle  of  incident  paraxial  rays  there  cor- 
responds a  homocentric  bundle  of  rays  refracted  at  the  first  surface. 
The  image-point  or  vertex  of  this  bundle  of  refracted  rays  may  be 
real  or  virtual;  but  in  either  case  it  is  to  be  regarded  as  lying  in  the 
second  medium,  even  though  the  actual  position  of  this  point  in  space 
may  lie  in  a  region  which  is  occupied  by  the  material  of  some  one  of 
the  other  media  (see  §  10).     This  bundle  of  rays  refracted  at  the  first 

174 


§  136.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


175 


surface  will  be  a  homocentric  bundle  of  paraxial  rays  incident  on  the 
second  surface,  to  which,  therefore,  there  corresponds  a  homocentric 
bundle  of  rays  refracted  at  this  latter  surface,  with  its  image-point 
lying  in  the  third  medium.  Proceeding  thus  from  surface  to  surface, 
remaining  always  a  bundle  of  homocentric  rays,  and  producing  a  point- 
image  in  each  successive  medium  of  the  series,  the  rays  emerge  finally 
into  the  last  medium  and  form  there  a  point-image,  which,  with  re- 
spect to  the  entire  centered  system  of  spherical  surfaces,  is  the  point 
conjugate  to  the  Object- Point  in  the  first  medium  from  which  the 
rays  originally  came.  Thus,  precisely  as  in  the  case  of  the  refraction 
of  paraxial  rays  at  a  single  spherical  surface,  we  have  also  for  the  re- 
fraction of  such  rays  through  a  centered  system  of  spherical  surfaces 
strict  collinear  correspondence  between  Object-Space  and  Image-Space. 
The  accompanying  figure  (Fig.  71)  represents  a  centered  system  of 
three  spherical  refracting  surfaces;  the  sections  of  the  surfaces  made 


FIG.  71. 

IMAGERY  BY  REFRACTION  OF  PARAXIAL  RAYS  THROUGH  A  CENTERED  SYSTEM  OF  SPHERICAL 
REFRACTING  SURFACES.  In  the  diagram,  the  spherical  surfaces  are  represented  by  the  straight 
lines  y\,  y%,  etc.;  in  the  figure  all  the  surfaces  are  represented  as  convex,  with  no  two  centres  C\ 
Cz,  etc.,  in  the  same  medium.  Moreover,  each  image  is  represented  as  a  real  image  formed  between 
the  centre  of  one  surface  and  the  vertex  of  the  next  following.  The  diagram  is  thus  drawn  merely 
for  the  sake  of  simplicity 


A\M\  =  u\,    A\M\'  = 


A\C\  =  r\, 


d\, 


'  =  «2'. 


=yz',    Mk'Qk'=ykr. 


by  a  plane  containing  the  optical  axis  (the  plane  of  the  diagram)  being 
shown  by  the  tangent-lines  ylt  y2,  etc.,  in  accordance  with  the  graphi- 
cal method  explained  in  §  113.  Consider  a  ray  MlBl  lying  in  the  plane 
of  the  diagram  which  crosses  the  optical  axis  at  the  point  designated 
by  Ml  and  meets  the  first  spherical  surface  at  the  incidence-point  Bl. 
After  refraction  at  this  surface  this  ray  crosses  the  axis  in  the  second 
medium  at  the  point  designated  by  M(,  which  is,  therefore,  the  axial 
image-point  in  this  medium  conjugate  to  the  Object-Point  M^  Inci- 
dent at  B2  on  the  second  surface,  the  ray  is  again  refracted,  and  again 
crosses  the  axis  at  a  point  M'2  which  is  the  image-point  in  the  third 


176  Geometrical  Optics,  Chapter  VI.  [  §  137, 

medium  conjugate  to  the  axial  Object-Point  Ml  in  the  first  medium. 
Any  one  of  these  image-points  may  be  real  or  virtual,  depending  on 
circumstances.  If  the  number  of  spherical  surfaces  is  w,  the  point  M'm 
where  the  ray  crosses  the  axis  after  refraction  at  the  last  surface  will 
be  the  image-point  which,  with  respect  to  the  entire  system  of  sur- 
faces, is  conjugate  to  the  axial  Object-Point  M^ 

The  diagram  shows  also  the  path  of  a  ray  which,  emanating  from  an 
Object- Point  Ql  near  the  optical  axis,  but  not  on  it,  traverses  the 
centered  system  of  spherical  surfaces.  The  actual  ray  whose  path  is 
drawn  is  the  ray  which  in  the  first  medium  is  directed  from  Q±  towards 
the  centre  Cl  of  the  first  surface,  and  which,  meeting  this  surface 
normally,  proceeds  into  the  second  medium  without  change  of  direc- 
tion; so  that  the  point  Q{  in  the  second  medium  which  is  conjugate 
to  Ql  must  lie,  therefore,  on  the  straight  line  connecting  Ql  and  CL. 
If  the  extra-axial  Object-Point  Ql  is  a  point  on  the  straight  line  per- 
pendicular to  the  optical  axis  at  Mlt  the  point  Q(  will  lie  on  the  straight 
line  perpendicular  to  the  optical  axis  at  M(,  and  the  straight  line  M{Q{ 
will  be  the  image  in  the  second  medium  of  the  short  Object-Line  MlQl 
in  the  first  medium.  The  image  of  Q[  produced  by  the  second  refrac- 
tion will  be  at  a  point  Q'2  in  the  third  medium,  which  is  the  point  of 
intersection  of  Q{  C2  with  the  perpendicular  to  the  optical  axis  at  M '2 ; 
thus,  M2Q2  will  be  the  image  in  the  third  medium  of  the  Object-Line 
MlQl.  The  point  Q'm  in  the  last  medium  will  be  the  Image-Point, 
with  respect  to  the  entire  system,  of  the  extra-axial  Object- Point  Qlt 
and  M'mQ'm  will  be  the  image,  produced  by  the  refraction  of  paraxial 
rays  through  a  centered  system  of  m  spherical  refracting  surfaces,  of 
a  small  Object-Line  MlQl  in  the  first  medium  perpendicular  to  the 
optical  axis. 

Thus,  exactly  as  in  the  case  of  a  single  spherical  refracting  surface, 
any  plane  of  the  Object-Space  perpendicular  to  the  optical  axis  of  a  centered 
system  of  spherical  refracting  surfaces  will  be  imaged  by  paraxial  rays 
by  a  plane  of  the  Image-Space  also  perpendicular  to  the  optical  axis. 

137.  The  abscissae,  with  respect  to  the  vertex  Ak  of  the  &th  sur- 
face, of  the  points  M'k_^  M'k  where  a  paraxial  ray  crosses  the  optical 
axis  before  and  after  refraction  at  this  surface  will  be  denoted  by 
%,  u'k,  respectively;  thus, 

AiM'k-\  =  «*,     AkM'k  =  uk, 

where  k  denotes  any  integer  from  i  to  m.  For  k  =  i,  we  have 
A^M^  =  «lf  since  we  write  here  Ml  instead  of  M'0.  The  radius  of  the 
kih  surface  is  denoted  by  r&,  and  is  defined  as  the  abscissa,  with  respect 


§  137.]  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  177 

to  the  vertex  Ak,  of  the  centre  Ck.  Moreover,  the  abscissa  of  the 
vertex  Ak+l  of  the  (k  +  i)th  surface  with  respect  to  the  vertex  Ak 
of  the  kth  surface,  called  the  thickness  of  the  (k  +  i)th  medium,  is 
denoted  by  dk.  Thus, 

•A-ifCk  =  rk,     AkAk+l  =  dk. 

Thus,  for  the  kth  Spherical  Refracting  Surface,  we  have,  according  to 
formula  (76): 


wherein,  since 

A^M't-L  +  Mk_lAk  =  A^Ak, 

the  value  of  uk  is  determined  by: 

**  =  «i-i  -<**-!•  (92) 

In  these  formulae  (91)  and  (92)  we  must  give  k  in  succession  all  integral 
values  from  k  =  i  to  k  =  m,  where  m  is  the  total  number  of  spherical 
surfaces  (Note  that  dQ  =  o).  Thus,  provided  we  know  the  magnitudes 
denoted  here  by  n,  r  and  d,  that  is,  provided  we  are  given  the  centered 
system  of  spherical  surfaces,  we  can,  by  means  of  these  recurrent 
formulae,  eliminate  in  order  the  magnitudes  denoted  by  u,  and  thus 
obtain  the  final  value  um  corresponding  to  a  given  value  of  ul  ;  that  is, 
determine  the  position  of  the  Image-Point  M'm  corresponding  to  a 
given  position  of  the  axial  Object-Point  M^. 

The  Focal  Point  E'  of  the  Image-Space  of  a  centered  system  of 
spherical  refracting  surfaces  is  the  point  where  a  paraxial  ray,  which 
in  the  first  medium  is  parallel  to  the  optical  axis,  crosses  this  axis  after 
refraction  at  the  last,  or  wth,  surface.  If  in  the  above  equations  we 
put  wx  =  oo,  then  um  =  AmE'  will  be  the  abscissa  of  the  point  E'  with 
respect  to  the  vertex  Am  of  the  last  spherical  surface.  We  shall  have 
(2m—  i  )  equations  with  (2m—  i  )  unknown  magnitudes,  viz.,  u2,  «3,-  •  -um 
and  «i,  «2»  '  '  '  um-  Accordingly,  by  successive  substitutions  we  can 
find  um.  Similarly,  the  Focal  Point  F  of  the  Object-Space  is  the  point 
where  a  ray  crosses  the  optical  axis  in  the  first  medium  which  emerges 
in  the  last  medium  parallel  to  the  optical  axis.  In  order  to  locate 
this  point  F,  we  must  put  um  =  <x>  and  find  the  value  of  the  abscissa 
«!  =  A1F  of  the  Focal  Point  F  with  respect  to  the  vertex  Al  of  the 
first  spherical  surface. 

13 


178  Geometrical  Optics,  Chapter  VI.  [  §  139. 

138.     The    Lateral    Magnification  F.      Putting    M'kQ'k  =  yk,    and 
making  use  of  formula  (85),  we  obtain  the  following  equations: 


yl       w  y        n2u2  yi        nkuk 

Multiplying  these  equations  together,  we  obtain: 

yu_  __  »!  u\'U2'  -  >uk 
y^        nk  u^  -  u2  -  -  -  uk 
The  ratio 

M'mQ'm      y'm 


is  called  the  Lateral  Magnification  of  the  centered  system  of  spherical 
surfaces  with  respect  to  the  axial  Object-Point  M^  Thus,  according 
to  the  formula  above,  we  have  : 

'  *  'k 

.-,        (93) 

where  the  symbol  II  means  the  continued  product  of  the  terms  ob- 
tained by  giving  k  in  succession  all  integral  values  from  k  =  I  to  k  =  m. 
Y  is  evidently  a  function  of  ulf 

139.  The  Principal  Points  of  a  Centered  System  of  Spherical 
Surfaces.  The  pair  of  conjugate  planes  perpendicular  to  the  optical 
axis  for  which  the  Lateral  Magnification  has  the  special  value  F  =  +  i  , 
so  that  for  this  pair  of  planes  object  and  image  are  equal  both  as  to 
magnitude  and  'sign,  were  called  by  GAUSS1  the  Principal  Planes  of 
the  optical  system;  and  the  two  conjugate  axial  points,  designated 
here  by  the  letters  A,  A',  where  the  Principal  Planes  were  cut  by 
the  optical  axis,  were  called  similarly  the  Principal  Points.  Putting 
Y  —  +  i  in  formula  (93),  we  have: 


-t 

£ti«,    *' 

which,  together  with  the  equations  (91)  and  (92),  gives  us  2m  equations 
with  2m  unknown  quantities,  whereby  we  can  determine  the  abscissae 
HI  =  A^A  and  um  =  AmA',  and  thus  ascertain  the  positions  of  the 
Principal  Points  A,  A'  . 

The  earlier  writers  on  Geometrical  Optics  proceeded  by  computing 
the  values  of  u,  u'  from  surface  to  surface.  MOEBIUS  and,  especially, 
GAUSS  strove  to  derive  general  formulae  for  finding  the  position  of 
the  image-point  conjugate  to  a  given  object-point,  without  involving 

1  C.  F.  GAUSS:  Dioptrische  Untersuchungen  (Goettingen,  1841),  p.  13. 


§  140.]  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  179 

the  tedious  process  of  tracing  the  path  of  the  ray  from  surface  to 
surface.  It  was  GAUSS  who  introduced  the  notion  of  the  so-called 
Cardinal  Points  of  the  optical  system.  These  are  certain  distinguished 
pairs  of  conjugate  axial  points,  the  most  important  of  which  are  the 
Principal  Points  A,  A',  which  are  briefly  referred  to  here.  We  may 
remark  that  GAUSS'S  method  marked  a  great  advance  in  the  science 
of  Geometrical  Optics,  and  led  to  very  simple  and  elegant  formulae. 
More  recently,  ABBE  (as  we  shall  see  in  the  following  chapter),  with- 
out employing  the  Cardinal  Points  at  all,  has  obtained  the  so-called 
"image-equations"  by  a  still  simpler  method  depending  only  on  the 
characteristics  of  the  Focal  Points  of  the  optical  system.  ABBE'S 
theory  of  optical  imagery  will  be  explained  at  length  in  the  following 
chapter;  where  will  be  found  also  a  more  extended  reference  to  the 
Cardinal  Points  of  the  system  (§  180). 

The  formulae  which  have  been  obtained  will  be  applied  in  this  chap- 
ter to  the  problem  of  the  refraction  of  paraxial  rays  through  an  infi- 
nitely thin  lens. 

ART.  40.     TYPES  OF  LENSES;   OPTICAL  CENTRE  OF  LENS. 

140.  A  centered  system  of  two  spherical  refracting  surfaces  con- 
stitutes what  is  known  in  Optics  as  a  Lens.  In  practice  the  Lens  is 
usually  surrounded  by  the  same  medium  on  both  sides,  and  we  shall 
assume  in  this  chapter  that  such  is  the  case.  We  may  denote  the 
indices  of  refraction  of  the  two  media  by  the  symbols  n  and  n' ',  thus, 

n  =  n^  =  n'2,     n  =  n(. 

Since  m  =  2,  we  obtain  from  equations  (91)  and  (92)  the  following 
formulae  for*  the  refraction  of  paraxial  rays  through  a  Lens  surrounded 
by  the  same  medium  on  both  sides: 


(94) 


U2  =  u(  -  d, 
n      n       n  —  n 
u2      u2          r2     ' 


where  here  we  use  d  instead  of  dl  to  denote  the  thickness  A^A2  of  the 
Lens.  Thus,  if  we  know  the  radii  rlt  r2  of  the  two  surfaces  of  the  Lens 
and  the  index  of  refraction  of  the  Lens-medium  relative  to  the  sur- 
rounding medium  (n'fn),  together  with  the  thickness  d  of  the  Lens, 
we  can  find  the  position  of  the  Image-Point  M2  conjugate  to  the  axial 


180 


Geometrical  Optics,  Chapter  VI. 


[§141. 


Object-Point  Mv  The  positions  of  the  Focal  Points  F  and  E'  may 
be  determined  by  putting,  first,  u'2  =  oo  and  solving  for  uv  —  A1F, 
and,  second,  WL  =  oo  and  solving  for  u'2  =  AJE! . 

The  Lateral  Magnification  with  respect  to  the  Object-Point  M-^  is 
obtained  at  once  by  putting  m  =  2  in  formula  (93) ;  thus,  we  have: 


y  =  ^2  = 


(95) 


141.     Lenses  may  be  conveniently  divided  into  two  main  classes, 
as  follows: 

(1)  Lenses  which  are  thickest  along  the  optical  axis.     In  this  group 
are  included,  therefore,  such  forms  of  lenses  as  are  shown  in  the  figure 
(Fig.  72),  viz.,  the  Bi-convex  Lens,  the  Plano-convex  Lens  and  the 
Convexo-concave  (or  Concavo-convex)  Lens  with  a  shallow  concavity 
(the  so-called  "Positive  Meniscus"). 

(2)  Lenses  which  are  thinnest  along  the  optical  axis.     To  this  group 
belong  the  Lenses  shown  in  Fig.  73,  viz.:  the  Bi-concave  Lens,  the 


c, 


FIG.  72  and  FIG.  73. 

TYPES  OF  I^ENSES.  In  Fig.  72  the  lenses  are  "  convergent"  or  positive  ;  in  Fig.  73  the  lenses  are 
"  divergent  "  or  negative  ;  assuming  that  the  lenses  are  glass  lenses  surrounded  by  air  and  not  too 
thick. 

A\At=d,    A\C\=r\, 


Plano-concave  Lens  and  the  Concavo-convex  (or  Convexo-concave) 
Lens  with  a  deep  concavity  (the  so-called  "Negative  Meniscus"). 

A  bundle  of  incident  parallel  paraxial  rays  falling  on  a  Lens  of  the 
first  group  —  supposed  to  be  a  moderately  thin  glass  lens  surrounded 
by  air  —  will  be  converged  to  a  real  focus  on  the  far  side  of  the  Lens; 
whereas,  under  the  same  circumstances,  a  beam  of  parallel  rays  will 
be  made  divergent  by  passing  through  a  Lens  of  the  second  group. 
On  account  of  this  characteristic  treatment  of  incident  parallel  rays, 
the  Lenses  of  the  first  group  are  sometimes  called  "Convergent" 


§  142.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


181 


Lenses,  and  those  of  the  second  group  are  called  "Divergent"  Lenses. 
But  this  property  depends  essentially  on  the  thickness  of  the  Lens 
and  on  the  relative  index  of  refraction. 

142.     Optical  Centre  of  Lens.     Any  ray,  whether  paraxial  or  not, 
which  leaves  the  Lens  (supposed  to  be  surrounded  by  the  same  medium 
on  both  sides)  in  a  direction  parallel  to  that  of  the  corresponding  inci-  \ 
dent  ray,  will  have  passed,  within  the  Lens,  (either  really  or  virtually)   j 
through  a  remarkable  point  on  the  optical  axis  called  the  Optical 
Centre  of  the  Lens.     In  order  to  prove  this  statement,  and  at  the 


FIG.  74. 

OPTICAL  CENTRE  OF  I<ENS  AT  THE  POINT  MARKED  O.  Any  ray  passing  through  O  emerges  from 
the  lens  in  a  direction  parallel  to  the  direction  of  the  incident  ray  ;  the  lens  being  surrounded  on 
both  sides  by  the  same  medium. 


A\C\  =  r\, 


d,     Z  C\B\O 


same  time  to  determine  the  position  of  this  point,  let  us  draw  through 
the  centres  Clt  C2  of  the  two  Lens-surfaces  any  two  parallel  radii 
C^B!,  C2B2  (Fig.  74)  :  then  the  point  O  where  the  straight  line  B^2 
crosses  the  optical  axis  is  a  fixed  point.  For  in  the  similar  triangles 
OC1B1  and  OC2B2  we  have: 

0,0 

C2O 
or 


C2B2 


C2O 


C2A 


or 


C2A 


C2A2 


182  Geometrical  Optics,  Chapter  VI.  [  §  143. 

and,  hence: 


AO      A2C2      r2 
And,  since 

A20  =  A2Al  +  Af)  =  A£)  -  AtA2  =  Af)  -  d, 

where  d  =  A±A2  denotes  the  thickness  of  the  Lens,  we  obtain  finally: 

AjO  =  — ^—  d.  (96) 

ri  r2 

Thus,  for  a  Lens  of  given  form  and  thickness,  this  equation  enables  us 
to  determine  the  abscissa,  with  respect  to  the  vertex  Al  of  the  first 
surface  of  the  Lens,  of  the  point  0,  which  is  a  fixed  point  on  the  optical 
axis,  since  its  position  is  independent  of  the  inclination  of  the  pair  of 
parallel  radii  ClBl  and  C2B2.  If,  therefore,  B1B2  represents  the  path 
of  a  ray  within  the  Lens  going  through  this  point  0,  the  directions  of 
the  corresponding  incident  and  emergent  rays  must  be  parallel,  since 
the  angle  of  refraction  a{  at  the  first  surface  is  equal  to  the  angle  of 
incidence  a2  at  the  second  surface.  The  optical  centre  0  will  be  recog- 
nized as  the  internal  centre  of  similitude  (or  perspective)  of  the  two 
circles  in  the  plane  of  the  diagram  which  have  Clt  C2  as  centres  and 
rlt  r2  as  radii,  respectively. 

In  the  figure,  as  drawn  here,  the  incident  ray  QBl  crosses  the  axis 
virtually  at  the  point  designated  by  N,  and  the  emergent  ray  B2Q' 
which  is  parallel  to  QBl  crosses  the  axis  virtually  at  the  point  desig- 
nated by  N'.  If  the  ray  is  a  paraxial  ray,  the  points  TV,  N'  will  be  a 
pair  of  axial  conjugate  points — the  so-called  "Nodal  Points"  of  the 
Lens. 

In  case  one  of  the  surfaces  of  the  Lens  is  plane,  the  optical  centre 
0  will  coincide  with  the  vertex  of  the  curved  surface,  as  is  evident 
from  formula  (96).  When  the  curvatures  of  the  two  surfaces  of  the 
Lens  have  the  same  sign,  as  is  the  case  with  either  the  positive  or 
negative  meniscus,  the  optical  centre  does  not  lie  within  the  Lens  at  all. 

ART.  41.     FORMULAE  FOR  THE  REFRACTION  OF  PARAXIAL  RAYS  THROUGH 
AN  INFINITELY  THIN  LENS. 

143.  When  the  Lens  is  so  thin  that  we  may  neglect  its  thickness 
(d)  in  comparison  with  the  other  linear  magnitudes  which  are  measured 
along  the  optical  axis,  we  have  the  case  of  an  Infinitely  Thin  Lens. 
In  comparison  with  the  other  dimensions  the  thickness  of  the  Lens  is 
often  quite  small,  but  an  Infinitely  Thin  Lens  is,  of  course,  unrealiz- 
able, so  that  such  a  Lens  is  sometimes  called  an  "ideal  Lens'l  If  we 


§  144.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


183 


put  AtA2  =  d  •  =  o,  this  is  equivalent  to  regarding  the  vertices  Alt  A2 
as  coincident,  and  the  Lens-surfaces  as,  therefore,  in  contact  with  each 
other.  The  approximate  formulae  that  are  obtained  under  these  cir- 
cumstances are  often  of  very  great  utility,  especially  in  the  preliminary 
design  of  an  optical  instrument  ;  and  in  many  cases  such  formulae  are 
quite  sufficient  to  enable  us  to  form  a  proper  idea  of  the  behaviour 
and  general  characteristics  of  a  real  Lens  of  not  too  great  thickness. 
144.  Conjugate  Axial  Points  in  the  case  of  the  Refraction  of  Paraxial 
Rays  through  an  Infinitely  Thin  Lens.  In  accordance  with  the  graphi- 
cal mode  of  representation  explained  in  §  113,  an  infinitely  thin  lens 
may  be  represented  in  a  diagram  by  a  straight  line  perpendicular  to 
the  optical  axis.  The  point  A  (Fig.  75)  where  this  straight  line  crosses 
the  axis  is  not  only  the  common  vertex  of  the  two  spherical  surfaces, 
but  it  is  also  the  position  of  the  optical  centre  of  the  Lens;  for,  ac- 
cording to  formula  (96)  ,  when  d  =  o,  the  optical  centre  coincides  with 
the  common  vertex  of 
the  Lens-surfaces.  The 
form  of  the  Lens  is 
shown  in  the  figure  by 
the  positions  of  the  cen- 
tres Clt  C2  of  the  two 
spherical  surfaces.  If 
in  the  second  of  formulae 
(94)  we  put  d  =  o,  we 
have  u2  =  u[.  Impos- 
ing this  condition,  and 
adding  the  two  other 
equations,  and  at  the 
same  time  writing  here 
u  and  u'  in  place  of  u^ 

and  u2,  respectively,  we  obtain  the  useful  abscissa-relation  for  the 
refraction  of  paraxial  rays  through  an  infinitely  thin  Lens  in  the  fol- 
lowing form  : 

'  ,     . 


FIG.  75. 

REFRACTION  OF  PARAXIAL  RAYS  THROUGH  INFINITELY 
THIN  lyENS.  M,  M'  are  a  pair  of  conjugate  axial  points. 
The  points  designated  in  the  diagram  by  M,  M' ,  Ci,  £2  and 
A  may  be  ranged  along  the  optical  axis  in  any  order  what- 
ever, depending  on  the  form  and  optical  properties  of  the 
lens  and  on  the  direction  of  the  incident  ray  MB.  The  lens 
represented  in  the  diagram  is  a  Biconvex  I^ens,  the  lens- 
medium  being  more  highly  refracting  than  the  surrounding 
medium. 

ACi  =  n,    ACz  =  n.    AM=u, 


n  —  n 


The  expression  on  the  right-hand  side  of  this  equation,  involving  only 
the  Lens-constants,  rlf  r2  and  n'  /n,  has  for  a  given  Lens  a  perfectly 
definite  value.  If  we  denote  this  constant  by  I  //,  so  that 


i  _  n'  —  n  /£       i\ 
7=    ~n~  \TI~  r 


(98) 


184  Geometrical  Optics,  Chapter  VI.  [  §  145. 

the  formula  above  may  be  written  as  follows: 

£-;-?•  (99) 

Thus,  having  determined  by  means  of  formula  (98)  the  value  of  the 
magnitude  denoted  by  /,  or  else  being  given  its  value  directly,  we  can 
ascertain  the  position  of  the  Image-Point  M'  corresponding  to  a  given 
axial  Object-Point  M ;  that  is,  knowing  u,  we  can  find  u1 ',  and  vice 
versa. 

It  may  be  remarked  that  equation  (99)  is  symmetrical  with  respect 
to  u  and  —  u' \  that  is,  if  —  u  be  written  in  place  of  u'  and  —  u'  in 
place  of  u,  the  equation  will  not  be  altered.  Hence,  if  the  Object-Point 
M  is  situated  on  the  axis  at  the  point  (u,  o)  and  the  Image-Point  Mf 
at  the  point  (u' ',  o),  and  if  the  Object- Point  is  then  supposed  to  be 
transferred  to  a  new  position  (—  u',  o),  the  new  Image-Point  will 
have  the  position  (  —  u,  o).  Or,  if  we  adjust  the  Lens  so  as  to  produce 
at  a  given  point  on  the  axis  the  image  of  a  fixed  Object-Point,  we  can 
find  two  positions  of  the  Lens  which  will  accomplish  the  purpose,  viz., 
a  position  for  which  the  Object- Point  has  the  abscissa  u  and  the  Image- 
Point  the  abscissa  u'  and  a  second  position  for  which  the  Object- 
Point  has  the  abscissa  —  u'  and  the  Image-Point  has  the  abscissa  —  u. 

145.  The  Focal  Points  of  an  Infinitely  Thin  Lens.  Putting  u  =  <*> 
in  formula  (99),  we  obtain: 

AE'=f, 

where  Ef  designates  the  position  on  the  optical  axis  of  the  Secondary 
Focal  Point  of  the  Infinitely  Thin  Lens.  Similarly,  putting  u'  —  oor 
we  find: 

AF=  -/, 

where  F  designates  the  position  of  the  Primary  Focal  Point  of  the  Lens. 
Thus,  the  two  Focal  Points  F  and  Ef  of  an  Infinitely  Thin  Lens  are 
equidistant  from  the  Lens,  and  on  opposite  sides  of  it. 

The  imagery  of  an  Infinitely  Thin  Lens  is  completely  determined 
so  soon  as  we  know  the  positions  of  the  three  points  A,  F  and  £'; 
and,  since  the  point  A  lies  midway  between  the  Focal  Points  F  and  £', 
Lenses  may  also  be  divided  into  two  classes,  as  follows: 

(i)  Lenses  in  which  the  points  F,  A,  Ef  are  ranged  along  the  optical 
axis  in  the  order  named  in  the  sense  in  which  the  light  is  propagated 
(therefore,  in  our  diagrams  from  left  to  right) ;  so  that  for  Lenses  of 
this  type  incident  rays  which  proceed  parallel  to  the  axis  will  be  con- 
verged to  a  real  focus  at  the  point  E'  beyond  the  Lens,  as  shown  in 


§  145.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


185 


the  first  diagram  of  Fig.  76:  and,  hence,  such  Lenses  are  called  Con- 
vergent Lenses.  They  are  also  called  Positive  Lenses,  because  FA  =  f 
is  positive,  if  we  take  the  direction  along  which  the  light  is  propagated 
as  the  positive  direction  of  the  ray.  Assuming  that  n'  >  n  (as,  for 


FIG.  76. 

Diagram  on  I^eft  represents  a  Convergent  L,ens  (/>  0)  ;  diagram  on  Right  represents  a  Divergent 
Iyens(/<0). 

f=FA,    e>  =  E'A<   f=  —  cf. 

example,  in  the  case  of  a  glass  lens  in  air),  the  sign  of/,  according  to 
formula  (98),  is  the  same  as  the  sign  of  (ijrl  —  i/r2).     In  the  Bicon- 


vex Lens  (rl  >  o,  r2  <  o),  the  Plano-convex  Lens 


=  oo,  r2  <  o, 


or  rl  >  o,  r2  =  oo)  and  the  Positive  Meniscus  (r2  >  rl  >  o)  —  that  is, 
for  all  Lenses  which  are  thicker  in  the  middle  than  towards  the  edges  — 
the  sign  of  (i/^  —  i/r2)  is  positive,  and,  therefore,/  >  o;  and,  hence, 
as  already  stated  (§  141),  such  lenses  (provided  n'  >  n)  are  convergent. 

(2)  Lenses  in  which  the  order  of  the  above-named  points  is  Er  ,  A,  F. 
For  lenses  of  this  class  incident  rays  which  proceed  parallel  to  the 
axis  are  made  divergent  by  passing  through  the  Lens,  and  emerge 
as  if  they  had  come  from  a  virtual  focus  at  the  Secondary  Focal  Point 
£',  lying  in  front  of  the  Lens,  as  shown  in  the  second  diagram  of 
Fig.  76.  Accordingly,  such  Lenses  are  called  Divergent  or  Negative 
Lenses,  since  here  FA  =  /  is  negative.  In  case  n'  >  n,  the  sign  of/,  as 
above  stated,  agrees  with  the  sign  of  (i/rl  —  i/r2).  In  the  Biconcave 
Lens  (TI  <  o,  r2  >  o),  the  Plano-concave  Lens  (rx  =  oo,  r2  >  o  or  rl  <  o, 
rz  =  oo  )  and  the  Negative  Meniscus  (r2  <  rl  <  o)  —  that  is,  for  all 
Lenses  which  are  thinner  in  the  middle  than  they  are  at  the  edges  — 
the  sign  of  (ijrl  —  i/r2)  is  negative;  and,  hence  (provided  n'  >  n), 
such  Lenses  are  divergent. 

Several  special  forms  of  Lenses  may  be  mentioned  here,  viz.  : 

The  Equiconvex  and  the  Equiconcave  Lens,  for  which  r2  =  —  rlf 
for  which  we  have,  therefore,  /  =  nr^^(n'  —  n).  In  the  case  of  the 
Equiconvex  Lens  r^  >  o,  and,  therefore  (assuming  n'  >  n),  we  have 
/  >  o;  whereas  for  the  Equiconcave  Lens  rl  <  o,  and,  therefore,/  <  o. 

The  Plano-convex  and  the  Plano-concave  Lens:  Assuming  that  the 
first  surface  of  the  Lens  is  the  plane  surface,  we  have  here  ^  =  00; 


186  Geometrical  Optics,  Chapter  VI.  [  §  146. 

so  that/  =  —  nr2j(nf  —  n).  The  sign  of  /  depends  therefore  on  the 
sign  of  r2. 

An  interesting  limiting  case  is  that  of  an  Ideal  Meniscus  in  which 
rl  =  r2.  Such  a  Lens  is  neither  thicker  nor  thinner  in  the  middle  than 
it  is  at  the  edges,  and,  therefore,  is  neither  convergent  nor  divergent. 
For  this  Lens  we  have/  =00,  and  therefore  also  u  =  u'.  Accordingly 
a  bundle  of  paraxial  rays  traversing  a  thin  Lens  of  this  description 
will  be  entirely  unaffected  so  far  as  changes  in  the  directions  of  the 
rays  are  concerned. 

146.  The  Focal  Lengths  /  and  er  of  an  Infinitely  Thin  Lens.  If 
the  Focal  Lengths  of  the  Lens  are  defined  exactly  in  the  same  way  as 
the  Focal  Lengths  of  a  single  spherical  refracting  surface  were  defined 
in  §  124,  we  shall  find  that  the  Focal  Lengths  of  an  Infinitely  Thin 
Lens  are  also  equal  to  FA  and  E'A  ;  that  is,  they  are  equal  to  the 
abscissae,  with  respect  to  the  Focal  Points  F  and  E'  ',  of  the  common 
vertex  A  of  the  two  Lens-surfaces.  If,  therefore,  we  denote  the  Focal 
Lengths  here  also  by/  and  e',  we  have: 

/=  FA,     e'  =  E'A; 

so  that  the  constant  /  introduced  above  and  defined  by  formula  (98)  , 
which  we  saw  was  equal  to  FA  ,  is  in  fact  the  Primary  Focal  Length  of 
the  Lens.  Obviously,  we  have  the  following  relations: 

/=_,'  =  —  -  ^  -  -.  (I00) 

(ri  -  n)  (r2  -  rj 

Thus,  the  focal  lengths  of  an  Infinitely  Thin  Lens  are  equal  in  magni- 
tude, but  opposite  in  sign.  If  we  reverse  a  thin  Lens,  so  that  the  first 
surface  of  the  Lens  is  the  surface  which  was  formerly  the  second  sur- 
face, we  do  not  alter  the  Focal  Lengths,  and  hence  the  character  of 
the  Lens  will  not  be  altered;  for  rt  becomes  —  r2,  and  the  formula 
above  is  not  altered. 

The  reciprocal  of  the  Focal  Length  is  called  the  Power  or  Strength 
of  the  Lens.  If  we  put  I//  =  <p,  and  if  also  the  curvatures  of  the  two 
surfaces  of  the  Lens  are  denoted  by  c  and  c',  that  is,  c  =  I  frlt  c'  =  I  /r2, 
formula  (98)  may  be  written  as  follows: 

n'  ~  n  /         t\  f       \ 

<p  =  —  -  —  (c-O;  (101) 


so  that  the  Power  of  an  Infinitely  Thin  Lens  is  proportional  to  the  dif- 
ference of  the  curvatures  of  the  two  surfaces  of  the  Lens. 


§  148.J  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  187 

Since  the  Focal  Lengths  of  an  Infinitely  Thin  Lens  are  equal  to 
the  distances  of  the  Lens  from  the  two  Focal  Points,  the  theory  of 
the  refraction  of  paraxial  rays  through  such  a  Lens  is  very  similar  to 
that  of  the  refraction  of  paraxial  rays  at  a  spherical  surface;  only,  in 
the  case  of  the  Lens  the  theory  is  simpler,  because  the  Focal  Points 
are  equidistant  from  the  Lens. 

147.  Putting  u2  =  u[  in  formula  (95),  and  writing  u,  u'  in  place 
of  ttlf  u'2,  respectively,  we  obtain  for  the  Lateral  Magnification  of  an 
Infinitely  Thin  Lens  : 

Y^  =  u~;  (,«) 

y        u 

that  is,  the  ratio  of  the  linear  dimensions  of  the  Object  and  Image  is  equal 
to  the  ratio  of  the  distances  of  the  Object  and  Image  from  the  Lens. 

If  x,  xf  denote  the  abscissae,  with  respect  to  the  Focal  Points  F,  E', 
of  the  conjugate  axial  points  M,  Mf,  respectively,  that  is,  if 

FM  =  x,     E'M'  =  x', 
then 

u  =  AM   =  AF  +    FM   =  x   -/, 
ur  =  AM'  =  AE'  +  E'M'  =  x'  -  e'\ 

and  substituting  these  values  in  formulae  (99)  and  (102),  we  obtain 
the  so-called  "Image-Equations"  of  an  Infinitely  Thin  Lens  in  the 
following  simple  and  convenient  forms  : 


The  abscissa-equation  is  the  same  as  the  characteristic  equation  of 
the  Central  Collineation  of  two  plane-fields  for  the  case  when  the 
invariant  c  =  +  i  (see  §  134).  It  may  be  derived  at  once  from  the 
projective  relation: 

(MAFE)  =  (M'AF'E'), 

where  E  and  Ff  are  the  infinitely  distant  points  of  the  two  correspond- 
ing ranges  of  Object-Points  and  Image-Points,  respectively,  lying  upon 
the  optical  axis  of  the  Lens. 

148.  Construction  of  the  Image  Formed  by  the  Refraction  of 
Paraxial  Rays  through  an  Infinitely  Thin  Lens.  In  the  diagrams 
(Figs.  77  and  78)  MQ  represents  a  very  short  Object-Line  perpen- 
dicular to  the  optical  axis  at  the  axial  Object-Point  M.  The  Infi- 
nitely Thin  Lens  is  itself  represented  by  the  straight  line  y  perpendicu- 
lar to  the  optical  axis  at  the  point  designated  by  A  .  Fig.  77  shows  the 


188 


Geometrical  Optics,  Chapter  VI. 


[  §  148. 


case  of  a  Convergent  Lens,  and  Fig.  78  shows  the  case  of  a  Divergent 
Lens.  Since  the  point  A  where  the  optical  axis  meets  the  Infinitely 
Thin  Lens  is  also  the  optical  centre  of  the  Lens  (  §144),  any  ray  directed 


M  E'     M'         .4rx.  f    »>        >/** 

Divergent  Lens 

FIG.  77  and  FIG.  78. 

REFRACTION  OF  PARAXIAL  RAYS  THROUGH  AN  INFINITELY  THIN  I^ENS.    Construction  of  Image. 
AM- 


u, 


-=AE'=f,  MQ=y, 


towards  A  will  emerge  from  the  Lens  without  change  of  direction, 
and,  hence,  the  straight  line  joining  any  pair  of  conjugate  points  <2, 
Q'  will  go  through  this  point  A.  Thus,  we  see  that  the  Object-Space 
and  Image-Space  of  an  Infinitely  Thin  Lens  are  in  perspective  relation 
to  each  other  with  respect  to  the  point  A  as  centre  of  perspective. 
This  is  obvious  also  from  formula  (102).  As  was  remarked  above 
(§  146),  the  imagery  in  the  case  of  an  Infinitely  Thin  Lens  is  quite 
similar  to  that  of  a  single  spherical  refracting  surface,  where  the  centre 
of  the  surface  is  the  centre  of  perspective  of  the  Object-Space  and 
Image-Space. 

Knowing  the  positions  of  the  axial  points  A  ,  F  and  E'  of  an  Infi- 
nitely Thin  Lens,  we  may  easily  construct  the  Image  M'Q'  conjugate 
to  M  Q.  All  that  we  have  to  do  is  to  locate  the  position  of  the  point 
Q',  and  then  draw  M'Q'  perpendicular  to  the  optical  axis  at  M'  .  The 
point  of  intersection  of  any  pair  of  emergent  rays  emanating  originally 
from  the  Object-Point  Q  will  suffice  to  determine  the  corresponding 
Image-Point  Q'  .  In  the  diagrams  (which  need  no  farther  explana- 
tion) three  such  rays  are  shown,  any  two  of  which  are  sufficient. 

The  imagery  in  the  case  of  the  Refraction  of  Paraxial  Rays  through 
an  Infinitely  Thin  Lens  is  exhibited  in  the  two  diagrams,  Figs.  79  and 
80,  the  first  of  which  shows  the  case  of  a  Convergent  Lens  and  the 


§  148.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


189 


second  the  case  of  a  Divergent  Lens.  The  numerals  I,  2,  3,  etc.,  desig- 
nate various  successive  positions  of  an  Object-Point,  which,  starting 
at  an  infinite  distance  in  front  of  the  Lens,  is  supposed  to  travel  towards 
the  Lens  along  a  straight  line  parallel  to  the  optical  axis.  The  cor- 
responding positions  of  the  Image-Point  on  the  straight  line  connect- 
ing the  point  V  with  the  Secondary  Focal  Point  E'  are  designated 
in  the  diagram  by  the  same  numerals  with  primes.  Thus  the  straight 
lines  n',  22',  etc.,  connecting  each  pair  of  conjugate  points,  will,  if 
they  are  drawn,  all  pass  through  the  perspective-centre  A.  In  both 
types  of  Lens  the  Object-Point  and  Image-Point  coincide  with  each 
other  at  the  point  V  on  the  Lens  itself,  and  hence  the  two  Principal 
Points  (§  139)  of  an  Infinitely  Thin  Lens  coincide  with  each  other  at 
the  point  A.  If  the  Object- Point  lies  beyond  the  Lens  (that  is,  to 
the  right  of  the  Lens  in  the  diagrams),  it  is  a  virtual  Object-Point. 
So  long  as  the  Object  is  in  front  of  the  Primary  Focal  Plane  of  a 
Convergent  Lens  (Fig.  79),  we  have  a  real,  inverted  Image  lying  on 


FIG.  79  and  FIG.  80. 

REFRACTION  OF  PARAXIAL  RAYS  THROUGH  INFINITELY  THIN  I^ENS.  IMAGERY  OF  IDEAL  I^ENS. 
The  numerals  1.2,3,  etc.,  show  a  number  of  selected  positions  of  an  object-point  supposed  to  move 
from  left  to  right  along  a  straight  line  parallel  to  the  optical  axis.  The  numerals  with  primes  show 
the  corresponding  positions  of  the  image-point  on  the  straight  line  E'  V. 

the  far  side  of  the  Secondary  Focal  Plane;  and  when  the  Object  is  in 
the  Primary  Focal  Plane,  the  Image  is  at  infinity.  If  the  Object  lies 
between  the  Primary  Focal  Plane  of  a  Convergent  Lens  and  the  Lens 


190  Geometrical  Optics,  Chapter  VI.  [  §  149. 

itself,  the  Image  is  virtual,  erect  and  magnified.  The  Image  of  a  virtual 
Object  formed  by  a  Convergent  Lens  is  real,  erect  and  diminished, 
and  lies  between  the  Lens  and  the  Secondary  Focal  Plane. 

In  a  Divergent  Lens  (Fig.  80).  the  Image  of  a  real  Object  is  always 
virtual,  erect  and  diminished,  and  lies  between  the  Secondary  Focal 
Plane  and  the  Lens.  A  Divergent  Lens,  however,  will  produce  a 
real  image  of  a  virtual  Object  which  is  placed  between  the  Lens  and 
the  Primary  Focal  Plane;  but  if  the  virtual  Object  lies  beyond  the 
Primary  Focal  Plane,  the  Image  produced  by  a  Divergent  Lens  will 
be  virtual  and  inverted  and  will  lie  in  front  of  the  Secondary  Focal 
Plane. 

When  u  =  —  u'  (that  is,  when  AM  —  M'A),  formula  (102)  shows  that 
Image  and  Object  are  equal  in  size  but  opposite  in  sign  (y'/y  =  —  i). 
Putting  u'  =  —  u  in  formula  (99),  we  find  u  =  —  2/  =  2^4F;  so  that 
for  this  special  position  of  Object  and  Image,  the  Primary  Focal 
Point  F  is  midway  between  the  points  designated  by  A  and  M. 
When  the  Object-Point  M  has  this  position,  the  Image-Point  Mf  is 
at  the  same  distance  from  the  Lens  on  the  other  side  of  it,  and  the 
Image  M'Q'  is  equal  to  QM.  If  the  point  M  moves  nearer  to  the 
Lens,  M'Q'  becomes  larger  than  QM,  and  if  the  point  M  moves  the 
other  way,  the  effect  will  be  exactly  opposite.  The  conjugate  axial 
points  M,  Mr  which  are  so  situated  with  respect  to  the  Lens  that 
AM  =  M'A  are  the  so-called  "Negative  Principal  Points"  of  the  Lens. 

149.  Refraction  of  Paraxial  Rays  through  a  Combination  of  Infi- 
nitely Thin  Lenses.  Suppose  we  have  a  centered  system  of  spherical 
surfaces  consisting  of  a  number  of  thin  lenses,  whose  optical  centres, 
ranged  along  the  optical  axis,  are  designated  by  Alt  A2,  etc.,  in  the 
order  named;  and  let 


denote  the  distance  between  the  kih  and  the  (k  +  i)th  Lenses  of  the 
system.  Moreover,  let  M'k  designate  the  point  where  a  paraxial  ray 
crosses  the  optical  axis  after  passing  through  the  &th  Lens,  and  let 

u   =  -4Af_f     u'  =  AM 


k         kk 

k 


denote  the  abscissae,  with  respect  to  Ak,  of  the  points  Mk-i,  M 
where  the  ray  crosses  the  optical  axis  before  and  after  passing  through 
the  &th  Lens.  Evidently,  we  shall  have  then  the  following  equations: 

*4-i  =  «*  +  d*-i,    7y  -  —  =  T  ;  (104) 

uk     uk    Jh 

where  fk  denotes  the  Primary  Focal  Length  of  the  kth  Lens.     If  the 


§  150.]  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  191 

number  of  Lenses  is  m,  we  must  give  k  in  succession  all  integral  values 
from  k  =  i  to  k  —  m\  noting  also  that  d0  =  o.  Instead  of  u'Q  we  write 
ul  =  AlMl,  where  Ml  designates  the  position  of  the  axial  Object-  Point. 
By  means  of  these  equations,  we  can  determine  the  abscissa  um  —  AmM'm 
of  the  Image-Point  M'm  corresponding  to  the  axial  Object-Point  Mlt 
provided  we  know  the  positions  and  Focal  Lengths  of  the  Lenses. 

The  Lateral  Magnification  of  a  combination  of  m  Infinitely  Thin 
Lenses  is  given  by  the  formula: 


In  the  special  case  when  the  Infinitely  Thin  Lenses  are  all  in  contact, 
so  that  we  have 

dl  =  d2  =  •  •  •  =  dk  =  o, 
we  obtain: 

%-i  =  %; 

and,  hence,  by  addition  of  all  the  equations  of  the  type 

iK-  i/%  =  i  /A, 

we  derive  the  following  formula  : 

k=m  -  - 

^     I  I  .  , 

.    ,-,  £/,-/•  (I06) 

The  combination  of  Thin  Lenses  in  Contact  is  therefore  seen  to  be 
equivalent  to  a  Single  Thin  Lens  of  Focal  Length  /  such  that  i  //  is 
equal  to  the  sum  of  the  reciprocals  of  the  Focal  Lengths  of  the  separate 
Lenses.  . 

ART.  42.     COTES'S    FORMULA    FOR    THE    "  APPARENT    DISTANCE  "    OF   AN 
OBJECT  VIEWED  THROUGH  ANY  NUMBER  OF  THIN  LENSES 

150.  More  than  twenty  years  ago,  Lord  RAYLEIGH1  directed  at- 
tention to  the  almost  forgotten  work  on  Optics  published  in  1783  by 
ROBERT  SMITH,2  professor  of  Astronomy  in  Cambridge  University 

'Lord  RAYLEIGH:  Notes,  chiefly  Historical,  on  some  Fundamental  Propositions  in 
Optics:  Phil.  Mag.,  (5),  xxi.  (1886),  466-476. 

2  ROBERT  SMITH:  A  Compleat  System  of  Opticks  in  four  books,  viz.  A  Popular,  a 
Mathematical,  a  Mechanical,  and  a  Philosophical  Treatise.  To  which  are  added  Remarks 
upon  the  Whole.  Cambridge,  1738.  This  work  is  in  two  large  octavo  volumes. 

A  German  translation,  edited  with  Notes  and  Additions  by  A.  G.  KAESTNER,  was 
published  by  RICHTER  in  Altenburg  in  1755.  The  title  of  this  translation  is:  Vollstaen- 
diger  Lehrbegriff  der  Optik  nach  Herrn  ROBERT  SMITHS  Englischen  mil  Aenderungen  und 
Zusaetzen  ausgearbeitet  -von  A.  G.  KAESTNER. 

In  1757  at  Avignon,  a  French  translation  by  LE  PERE  PEZENAS  was  published  with  the 


192  Geometrical  Optics,  Chapter  VI.  [  §  151. 

and  afterwards  Master  of  Trinity  College.  SMITH  was  the  literary 
executor  of  his  cousin  ROGER  COTES,  the  first  Plumian  Professor  of 
Astronomy  in  Cambridge  University  and  the  editor  of  the  second 
edition  of  NEWTON'S  Principia,  who  died  in  1716,  <zt.  34,  leaving  un- 
finished a  series  of  elaborate  researches  in  Optics.  "Had  COTES  lived, 
we  might  have  known  something!"  is  a  speech  attributed  to  NEWTON, 
with  whom  he  was  closely  associated.  Chapter  V  of  Book  II  of 
SMITH'S  treatise  is  founded  on  a  "noble  and  beautiful  theorem",  said 
to  have  been  the  last  piece  of  work  of  COTES'S  life.  The  theorem  is 
remarkable  as  being,  perhaps,  the  earliest  generalization  in  Optics, 
and  it  was  used  by  SMITH  in  a  masterly  fashion  in  deriving  a  number 
of  very  important  corollaries,  which,  as  RAYLEIGH  observes,  were 
afterwards  "rediscovered  in  a  somewhat  different  form  by  LAGRANGE, 
KIRCHHOFF,  and  VON  HELMHOLTZ".  For  these  reasons,  and,  also, 
on  account  of  its  elegant  form  and  intrinsic  value,  COTES'S  Theorem 
should  not  be  omitted  from  a  treatise  on  Optics — especially  too  as  it 
is  now  very  hard  to  obtain  a  copy  of  SMITH'S  Optics.  The  theorem 
is  given  by  P.  CULMANN  in  his  article  on  Die  Realisierung  der  optischen 
Abbildung,  published  in  the  first  volume  of  Die  Theorie  der  optischen 
Instrumente,  edited  by  M.  VON  ROHR  (Berlin,  1904). 

151.  COTES  used  the  term  "Apparent  Distance"  to  mean  the  dis- 
tance at  which  the  Object  would  have  to  be  placed  so  as  to  appear  to 
the  naked  eye  of  the  same  angular  magnitude  as  its  Image  appeared 
when  viewed  through  the  optical  system.  Although  this  meaning  has 
never  come  into  general  use,  the  term  will  be  employed  in  this  sense 
in  the  following  exposition  of  COTES'S  Theorem. 

In  Fig.  8 1  let  MQ  represent  an  Object-Line  perpendicular  to  the 
optical  axis  of  a  system  of  Infinitely  Thin  Lenses  whose  optical  centres 
are  at  the  points  designated  by  Alt  A2,  etc.  In  the  diagram  the  sys- 
tem is  represented  as  consisting  of  only  three  lenses.  Let  the  broken 
line  QP^P.^P^  represent  the  path  of  the  ray  proceeding  from  the 
end-point  Q  of  the  Object  and  entering  the  eye  supposed  to  be  placed 
on  the  axis  at  the  point  L's.  The  points  designated  by  Plf  P2,  P3 
are  the  points  where  this  ray  meets  the  lenses  A},  A2,  A3,  respectively. 
The  point  LL  is  the  point  where  the  incident  ray  QPl  crosses  the  optical 
axis,  and  the  point  L'&  is  the  point  where  the  ray  crosses  the  axis  after 

title:  Cours  complet  d'optique,  traduit  de  I'anglois  de  ROBERT  SMITH,  contenant  la  theorie, 
la  pratique  et  les  usages  de  cette  science.  Avec  des  additions  considerables  sur  toutes  les  nou- 
velles  decouvertes  qu'on  a  faites  en  cette  maniere  depuis  la  publication  de  I'ouvrage  anglois. 
RAYLEIGH  mentions  another  French  translation  also  published  in  1767,  at  Brest,  by 
DUVAL  LEROY. 


§  151.] 


Refraction  of  Paraxial  Rays  Through  a  Thin  Lens. 


193 


passing  through  all  three  of  the  lenses.  In  the  diagram  the  inter- 
section at  L!  is  represented  as  virtual,  and  that  at  L'3  is  real.  In  the 
same  way  L(  and  L'2  designate  the  positions  of  the  points  where  the 


FIGURE  USED  IN  DEDUCING  COTES'S  THEOREM.  A\P\,  AzPz,  AsPs  represent  three  infinitely  thin 
lenses.  In  the  diagram  these  lenses  are  all  represented  as  concave  or  divergent  lenses.  MQ  is 
object-line  perpendicular  to  optical  axis  of  system  of  lenses.  QP\P*PzU  is  the  outermost  ray  pro- 
ceeding from  the  end-point  Q  of  the  object  and  traversing  all  the  lenses.  The  eye  is  supposed  to 
be  placed  on  the  axis  at  L$' '.  KLj  is  the  "  apparent  distance  "  of  the  object  from  the  eye. 

ray  crosses  the  axis  after  passing  through  the  first  and  second  lenses, 
respectively:  both  of  these  intersections,  as  shown  in  the  diagram,  are 
virtual.  From  the  points  Q,  Pl  and  P2  draw  straight  lines  parallel 
to  the  optical  axis  and  produce  them  until  they  meet  the  straight  line 
determined  by  the  emergent  ray  PZL\,  and  from  each  of  these  points 
of  intersection  let  fall  perpendiculars  on  the  axis  at  the  points  desig- 
nated by  K,  J  and  H,  respectively. 

The  "apparent  distance"  of  A2P2,  regarded  as  an  object  viewed 
through  the  Lens  As  is  HL^'t  and  from  the  similar  right  triangles 
of  the  figure  we  obtain  the  following  proportions: 


A2P2 


and  consequently: 


Since  L'2,  L'3  are  a  pair  of  conjugate  axial  points  with  respect  to  the 
Lens  AB,  we  have,  according  to  formula  (99) : 


where  /3  denotes  the  Focal  Length  of  the  Lens  As.     Introducing  this 

14 


194  Geometrical  Optics,  Chapter  VI.  [  §  151. 

value  of  i/A^L2  in  the  above,  we  obtain: 


A   A  . 

^2^3 


f 
/3 

In  the  same  way,  the  "apparent  distance"  of  AlPl  regarded  as  an 
Object  viewed  through  the  Lenses  A2  and  A3  is  JL3.  Here  we  have 
the  proportions: 

J  J-'          **-^-  ^-^-J 


or 


In  the  same  way,  also,  we  have  here : 
i  ill 


and  hence: 

f  _     A   -r  f        A.  i  A.  2  •  A  2Lt3       sLlA3'  A$Li$       A1A 
3=^3~  /,  /a 

Again,  the  "apparent  distance"  of  the  Object-Line  MQ  viewed 
through  the  three  Lenses  A^  A2  and  Az  is  KL^-  and,  as  before: 

MQ       ML, 


or 


Also, 

i  i         j  _  HL'3  (    i          i\       i_ 

IA  ~  T&  ~7,  ~  TL;  v^T;  ~7J  ~/3 


so  that 

XL;  =  /L;  +  MA,(I  -  ^  -  ^  - 

\  /3  J2  Jl    / 

and,  finally  we  obtain  the  following  formula  for  the  "apparent  dis- 


§  152.]  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  195 

tance"  of  the  Object  viewed  through  the  Lenses  Alt  A2  and  Az: 
A^      MA2-A2L', 


MA,  •  A,A2' 

/,                                      /, 

•^2-^3    I    MA,  '  A.,A3  '  A. 

/3 

3l4      MA2-A2AZ-A 

UL; 

/I  /I 

/./, 

JJt 

L*  -^1-^2'  ^•|~s/  A^L,. 

'.  (io7) 

/,/,/, 

This  is  COTES'S  Formula  for  the  case  when  the  system  is  composed 
of  three  lenses  Alt  A2,  A3\  but  the  law  of  the  formation  of  the  terms 
is  apparent,  and  the  formula  can  be  immediately  written  for  a  system 
of  any  number  of  Lenses.  Thus,  if  we  observe  that  the  piece  of  the 
optical  axis  included  between  the  Object  at  M  and  the  eye  at  L'3 
may  be  considered  as  divided  by  the  Lenses  at  Alt  A2,  Az  into  two, 
three  and  four  segments  in  the  following  ways  : 


ML'3  =  MA,  +  A,L'Z  =  MA2  +  A^  =  MA3  +  AZL'Z 
=  MAl  +  A,A2  +  A2L'Z  =  MA,  +  A,AB  +  ABL'3 
=  MA2  +  A2A,  +  A,L',  =  MA,  +  A,A2  +  A2AB 


it  will  be  seen  that  the  members  of  each  of  these  groups  when  multi- 
plied together  form  the  products  which  are  the  numerators  of  the 
fractions  on  the  right-hand  side  of  equation  (107),  while  the  denomi- 
nators are  the  products  of  the  Focal  Lengths  of  the  Lenses  which 
occur  in  the  numerators;  the  signs  of  the  fractions  being  positive  or 
negative  according  as  the  number  of  factors  in  the  denominator  is 
even  or  odd.  The  "apparent  distance"  is  equal  to  the  real  distance 
added  to  the  algebraic  sum  of  the  set  of  fractions  whose  numerators 
and  denominators  are  formed  according  to  the  rule  just  explained. 

A  general  proof  of  COTES'S  Theorem  was  given  by  LAGRANGE/ 
who  was  evidently  acquainted  with  SMITH'S  work  on  Optics,  as  he 
refers  to  it  in  his  paper. 

152.  Lord  RAYLEIGH  in  the  article  above-mentioned  (§  150)  quotes 
at  length  several  of  the  corollaries  which  SMITH  derives  from  COTES'S 
Theorem,  the  first  of  which  is  as  follows: 

"While  the  glasses  are  fixt,  if  the  eye  and  object  be  supposed  to 
change  places,  the  apparent  distance,  magnitude  and  situation  of  the 
object  will  be  the  same  as  before.  For  the  interval  M  L'3  being  the 

1  J.  L.  DE  LAGRANGE:  Sur  la  theorie  des  lunettes:  M'emoires  de  I'Acad.  de  Berlin  (1780), 
162-180. 


196  Geometrical  Optics,  Chapter  VI.  [  §  152. 

same,  and  being  divided  by  the  same  glasses  into  the  same  parts, 
will  give  the  same  theorem  for  the  apparent  distance  as  before." 

Thus,  in  Fig.  81,  if  we  suppose  that  the  axial  point  of  the  Object 
is  at  L3  and  that  the  centre  of  the  pupil  of  the  eye  is  on  the  axis  at 
the  point  designated  by  M,  then  A3P3  will  be  proportional  to  the 
breadth  at  the  object-glass  A3  of  the  bundle  of  incident  rays  from  the 
axial  Object- Point  L3,  and  MQ  will  be  proportional  to  the  breadth  of 
the  corresponding  bundle  of  emergent  rays  where  they  enter  the  eye 
at  M,  and  from  the  figure  we  have  evidently: 

MQ  __  KL3 
A3P3      A3L$ 

whence  is  derived  SMITH'S  Second  Corollary,  which  he  states  as  fol- 
lows: 

"When  an  object  MQ  is  seen  through  any  number  of  glasses,  the 
breadth  of  the  principal  pencil  where  it  falls  on  the  eye  at  L3,  is  to 
its  breadth  at  the  object-glass  Alt  as  the  apparent  distance  of  the 
object,  to  its  real  distance  from  the  object-glass;  and  consequently 
in  Telescopes,  as  the  true  magnitude  of  the  object,  to  the  apparent." 

This  very  striking  result  can  be  put  in  a  different  form.  Thus, 
from  the  figure,  we  obtain: 


A3P 
and  therefore: 


The  expression  on  the  right-hand  side  of  this  equation,  according  to 
formula  (105),  is  the  value  of  the  Lateral  Magnification  y'3/yl  at  the 
conjugate  axial  points  Llt  L3,  so  that  we  have: 

y'3 


MQ       M       A,P,   A2Pt_ML1   A,L{   A2L'2 
AZPB      A^LH   i 

.L/I  *  ^±2 -"2  *  "3-"3 


ML,      yi~ 
Moreover, 

KL't  _  tan  Z.A1L1P1  _  tan0l 
ML^  ~  tan  Z  A3L3P3  ~  tan  03  ' 

and  hence  we  derive  the  formula  : 

»  3/3  •  tan  #3  =  yl  -  tan  6^  ; 


§  152.]  Refraction  of  Paraxial  Rays  Through  a  Thin  Lens.  197 

or,  if  the  system  consists  of  m  Lenses: 

y'm'tan8'm  =  y1-tan01.  (108) 

This  formula,  which  was  given  by  LAGRANGE1  more  than  fifty  years 
after  the  publication  of  SMITH'S  Optics,  is  a  particular  case  of  the  gen- 
eral formula  usually  known  in  Optics  as  the  HELMHOLTZ  Equation 
(see  §  194). 

1  J.  L.  DE  LAGRANGE:  Sur  une  loi  generate  d'optique:  Memoires  de  I'Acad.  de  Berlin, 
1803. 


CHAPTER  VII. 

THE  GEOMETRICAL  THEORY  OF  OPTICAL  IMAGERY. 

I.     INTRODUCTION. 
ART.  43.     ABBE'S  THEORY   OF   OPTICAL  IMAGERY. 

153.  The  function  of  an  optical  instrument  is  to  produce  an  image 
of  an  external  object.     Each  point  of  the  object  is  the  base  (or  vertex) 
of  a  bundle  of  rays,  of  which,  in  general,  only  a  part  is  utilized  in  the 
formation  of  the  image.     These  object-rays  which  are  affected  by  the 
instrument  are  called  the  "incident"  rays.     Within  the  apparatus  these 
rays  undergo  a  series  of  refractions  (or  reflexions)  at  the  plane  or  curved 
boundary-surfaces  of  suitably  disposed  optical  media;  and,  thus  modi- 
fied, they  "emerge"  into  the  last  medium  and  form  there  a  more  or 
less  perfect  image  of  the  object,  which  may  be  "real"  or  "virtual", 
etc. ;  the  nature  of  the  image  in  the  several  respects  of  position,  dimen- 
sions, orientation,  etc.,  depending  primarily  on  the  peculiarity  and 
design  of  the  instrument  itself.     Proceeding  from  any  point  P  of  the 
Object,  a  bundle  of  incident  rays  "enters"  the  optical  instrument, 
and  emerging  therefrom,  a  portion  of  these  rays  at  least,  if  not  all  of 
them,  will  intersect  ("really"  or  "virtually")  in  the  corresponding,  or 
"conjugate",  point  Pf  of  the  Image.     In  the  case  of  an  ideal,  or  geo- 
metrically perfect,  image,  all  of  the  emergent  rays  corresponding  to 
the  rays  of  the  bundle  of  incident  rays  P  will  intersect  in  the  Image- 
Point  P' ;  so  that  a  homocentric  bundle  of  object-rays  will  be  (as  the 
German  writers  say)  "imaged"  (abgebildet)  by  a  homocentric  bundle 
of  image-rays. 

154.  Until  comparatively  recent  times  the  method  of  investigation 
of  the  relations  between  image  and  object  in  Optics  was  to  advance 
by  a  process  of  mathematical  induction  from  simple  special  cases  to 
more  complex  general  cases  of  homocentric  imagery.     This  method 
was  used  with  conspicuous  success  by  ROGER  COTES   (§  150),  first 
Plumian  Professor  of  Astronomy  in   Cambridge  University,  whose 
brilliant  and  original  contributions  to  optical  science  were  cut  short 
by  his  untimely  death  (1716)  at  the  age  of  thirty-four  years.     The 
same  method  was  employed  also  by  C.  F.  GAUSS  in  his  famous  Diop- 
trische  Untersuchungen  (Goettingen,  1841),  who  developed  completely 
the  theory  of  the  refraction  of  paraxial  rays  through  a  centered  sys- 

198 


§  155.]  The  Geometrical  Theory  of  Optical  Imagery.  199 

tern  of  infinitely  thin  lenses.  By  substituting  in  place  of  the  original 
data,  such  as  the  radii,  refractive  indices,  etc.,  certain  constants  of  a 
much  more  general  kind,  GAUSS  obtained  remarkably  simple  formulae, 
which  marked  a  great  advance  in  optical  theory  and  added  a  new 
encouragement  to  such  investigations.  But  even  GAUSS,  with  his 
extraordinary  insight  and  rare  gift  of  analysis,  seems  not  to  have  dis- 
cerned that  the  general  laws  of  optical  imagery  are  independent  of  all 
special  assumptions  as  to  the  particular  mode  of  producing  the  image. 

MoEBius,1  indeed,  came  nearer  to  the  real  and  essential  idea  of 
an  optical  image  when  he  pointed  out  that  the  unique  connection  be- 
tween Object-Point  and  Image-Point  in  the  case  of  the  refraction  of 
paraxial  rays  at  a  spherical  surface  is  equivalent  to  the  expression 
of  the  relation  of  Collinear  Correspondence  between  Object-Space  and 
Image-Space;  and  that  if  this  is  true  in  the  case  of  a  single  spherical 
refracting  surface,  it  must  be  true  also  for  the  relation  between  object 
and  image  in  the  refraction  of  paraxial  rays  through  a  centered  system 
of  spherical  refracting  surfaces;  and,  hence,  finally,  that  all  the  formulae 
showing  the  relation  between  object  and  image  in  such  a  case  as  this 
were  deducible  from  the  theory  of  Collinear  Correspondence.  This 
presentment  was  quickly  seized  by  other  investigators  (as  F.  LiPPicn,2 
A.  BECKS  and  H.  HANKEI/)  who,  following  the  lead  of  MOEBIUS,  and, 
like  him,  employing  the  methods  of  projective  geometry,  extended 
this  idea  of  optical  imagery  to  less  simple  cases.  Thus,  for  example, 
F.  LiPPiCH5  showed  that  there  is  also  collinear  correspondence  of  ob- 
ject and  image  in  the  case  of  infinitely  narrow  bundles  of  rays  incident 
on  a  spherical  refracting  surface  at  finite  slopes.  Yet  neither  MOEBIUS 
himself  nor  any  of  his  followers  in  this  mode  of  treating  the  matter 
was  able  to  discard  entirely  the  idea  that  some  kind  of  Dioptric  action 
was  essential  for  the  production  of  an  optical  image.  At  least  not  one 
of  them  stated  distinctly  that  a  purely  geometrical  assumption  was  all 
that  was  necessary,  viz.,  that  an  optical  image  is  produced  by  rays.\ 

155.  A  remarkable  paper  "On  the  General  Laws  of  Optical  Instru- 
ments" was  contributed  in  1858  by  JAMES  CLERK  MAXWELL  to  The 

1  A.  F.  MOEBIUS:  Entwickelung  der  Lehre  von  dioptrischen  Bildern  mit  Huelfe  der 
Collineations- Verwandschaf t :   Leipziger  Berichte,  vii.  (1855),  8-32. 

2  F.  LIPPICH:  Fundamentalpunkte  eines  Systemes  centrirter  brechender  Kugelflaechen: 
Mfiltheilungen  des  naturwissenschaftlichen  Vereines  fur  Steiermark,  ii.  (1871),  429—459. 

3  A.  BECK:  Die  Fundamentaleigenschaften  der  Linsensysteme  in  geometrischer  Dar- 
stellung:   Zft.  f.  Math.  u.  Phys.,  xviii.  (1873),  588-600. 

4H.  HANKEL:  Die  Elemente  der  projektivischen  Geometric  in  synthetischer  Behandlung 
(Leipzig,  1875). 

5F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme  an 
Kugelflaechen:  Wiener  Denkschr.,  xxxviii.  (1878),  163-192. 


200  Geometrical  Optics,  Chapter  VII.  [  §  155. 

Quarterly  Journal  of  Pure  and  Applied  Mathematics,  ii.,  233-246;  it 
is  reprinted  in  the  collection  of  MAXWELL'S  Scientific  Papers,  vol.  i., 
271-285.  In  an  introduction  to  this  article,  MAXWELL  describes  the 
undertaking  as  follows: 

"The  investigations  which  I  now  offer  are  intended  to  show  how 
simple  and  how  general  the  theory  of  optical  instruments  may  be  ren- 
dered, by  considering  the  optical  effects  of  the  entire  instrument,  with- 
out examining  the  mechanism  by  which  these  effects  are  obtained.  I 
have  thus  established  a  theory  of  'perfect  instruments',  geometrically 
complete  in  itself,  although  I  have  also  shown  that  no  instrument 
depending  on  refraction  and  reflexion  (except  the  plane  mirror)  can 
be  optically  perfect." 

A  "perfect  instrument"  is  one  which  is  free  from  "certain  defects 
incident  to  optical  instruments";  thus,  according  to  MAXWELL,  "a 
perfect  instrument  must  fulfil  three  conditions: 

"I.  Every  ray  of  the  pencil,  proceeding  from  a  single  point  of  the 
object,  must,  after  passing  through  the  instrument,  converge  to,  or 
diverge  from,  a  single  point  of  the  image.  The  corresponding  defect 
when  the  emergent  rays  have  not  a  common  focus,  has  been  appro- 
priately called  (by  Dr.  WHEWELL)  Astigmatism. 

"II.  If  the  object  is  a  plane  surface,  perpendicular  to  the  axis  of 
the  instrument,  the  image  of  any  point  of  it  must  lie  in  a  plane  perpen- 
dicular to  the  axis.  When  the  points  of  the  image  lie  in  a  curved 
surface,  it  is  said  to  have  the  defect  of  curvature. 

"III.  The  image  of  an  object  on  this  plane  must  be  similar  to  the 
object,  whether  its  linear  dimensions  be  altered  or  not;  when  the 
image  is  not  similar  to  the  object,  it  is  said  to  be  distorted." 

Assuming  that  the  image  is  free  from  these  three  defects,  and  is 
therefore  a  "perfect  image",  MAXWELL  derives  formulae  for  the  rela- 
tive positions  and  magnitudes  of  the  object  and  image  which  are  pre- 
cisely equivalent  to  the  formulae  obtained  by  GAUSS  ;  but  the  difference 
consists  in  the  fact  that,  whereas  GAUSS'S  investigations  are  based 
on  certain  physical  assumptions  not  only  in  regard  to  the  Law  of 
Refraction  of  light-rays,  but  also  as  to  a  centered  system  of  spherical 
surfaces  and  paraxial  rays,  the  modus  operandi  is  left  out  of  consider- 
ation entirely  by  MAXWELL,  who  shows  that  an  optical  image,  how- 
ever it  may  be  produced,  provided  it  is  free  from  the  geometrical 
"defects"  above  enumerated,  must  have  certain  perfectly  definite 
geometrical  relations  with  the  object.  This  very  important  idea  seems 
to  have  been  clearly  perceived  and  distinctly  stated  by  MAXWELL 
first  of  all. 


§  157.]  The  Geometrical  Theory  of  Optical  Imagery.  201 

156.  The  most  notable  contribution  in  recent  years  to  the  litera- 
ture of  Geometrical  Optics  is  Dr.  S.  CZAPSKI'S  Theorie  der  optischen 
Instrumente  nach  ABBE,  the  first  edition  of  which  was  published  in 
Breslau  in  1893.  In  this  brilliant  work,  recognized  immediately  as 
an  epoch-making  book,  was  set  forth  for  the  first  time  a  complete  and 
masterly  exposition  of  the  remarkable  theories  of  Professor  ABBE,  of 
Jena. 

ABBE,  without  a  knowledge  of  the  investigations  of  MOEBIUS  and 
MAXWELL,  discerned  even  more  clearly  than  they  that  the  physical 
agency  or  mechanism  which  was  employed  in  the  actual  formation 
of  an  optical  image  was  in  no  wise  involved  in  the  geometrical  theory 
of  optical  imagery ;  so  that  without  any  special  assumptions  whatever 
as  to  the  construction  or  constitution  of  the  optical  apparatus,  and 
even  without  reference  to  the  physical  laws  of  reflexion  and  refraction, 
he  deduced  in  the  simplest  and  most  direct  way  all  the  laws  concerning 
the  relative  positions,  dimensions,  etc.  of  the  object  and  image. 

Thus,  the  fundamental  and  essential  characteristic  of  optical  im- 
agery is  a  point-to-point  correspondence,  by  means  of  rectilinear  rays, 
between  object  and  image;  and  from  this  one  assumption — at  once 
the  most  natural  and  the  most  obvious — ABBE,  in  his  celebrated 
university  lectures,  used  to  deduce  the  general  laws  of  optical  images. 

The  advantage  of  this  is  that  in  investigating  an  actual  image  pro- 
duced by  an  optical  instrument  it  will  be  possible  to  separate  what 
in  the  laws  of  this  image  depends  on  the  general  fundamental  laws  of 
optical  imagery  and  what  is  due  to  the  particular  mode  of  producing 
the  image.  Moreover,  although  to-day  a  certain  optical  instrument 
may  be  a  mechanical  impossibility,  it  is  possible  to  say  whether  such 
a  system  is  theoretically  practicable;  so  that  the  geometrical  theory 
will  point  the  way  of  future  inventions. 

In  the  modern  geometry  this  unique  point-to-point  correspondence 
by  means  of  rectilinear  rays  between  image  and  object  is  called  "  Col- 
lineation" — a  term  introduced  by  MOEBIUS  in  his  great  work  entitled 
Der  barycentrische  Calcul  (Leipzig,  1827). 

II.  THE  THEORY  OF  COLLINEATION,  WITH  SPECIAL  REFERENCE  TO  ITS 
APPLICATIONS  TO  GEOMETRICAL  OPTICS. 

ART.  44.     TWO  COLLINEAR  PLANE-FIELDS. 

157.  Definitions.  In  this  treatment  we  shall  employ  the  beauti- 
ful and  appropriate  methods  of  projective  geometry.  As  some  readers 
may  not  be  entirely  familiar  with  the  terms  here  employed,  a  brief 
introduction  may  be  required. 


202  Geometrical  Optics,  Chapter  VII.  [  §  158. 

The  totality  of  points  and  straight  lines  which  are  contained  in  a 
plane  is  called  a  "plane-field",  and  the  plane  is  then  said  to  be  the 
"base  "of  this  system  of  points  and  lines.  All  the  points  lying  on  a 
straight  line  of  the  field,  taken  together,  form  a  "range  of  points" ', 
the  straight  line  itself  being  called  the  "base"  of  the  point-range.  A 
straight  line  considered  as  a  whole  (that  is,  without  reference  to  the 
points  which  lie  on  it)  is  called  a  "ray" '.  All  the  straight  lines  of  a 
plane-field  which  go  through  one  point  form  a  "pencil  of  rays" ,  and 
the  common  point  of  intersection  of  these  rays  may  be  regarded  as 
the  "base"  of  the  pencil. 

Two  plane-fields  TT  and  IT'  are  said  to  be  collinear,  if  to  every  point  P 
of  TT  there  corresponds  one  point  P'  of  IT',  and  to  every  straight  line  p 
of  TT  which  goes  through  P  there  corresponds  a  straight  line  p'  of  irf  which 
goes  through  P'. 

The  totality  of  rays  which  go  through  a  single  point  0  in  space 
is  called  a  "bundle  of  rays" ,  so  that  a  bundle  of  rays  consists  of  an 
infinite  number  of  pencils  of  rays.  We  speak  also  of  a  "bundle  of 
planes" ,  meaning  thereby  the  totality  of  planes  which  pass  through 
one  point  0.  In  either  case  the  point  0  which  is  common  to  all  the 
elements  of  the  bundle  is  the  "base"  of  the  bundle.  A  "sheaf  of 
planes"  is  the  term  applied  to  the  totality  of  planes  which  all  have  one 
common  line  of  intersection :  thus,  in  a  bundle  of  planes  are  comprised 
an  infinite  number  of  sheaves  of  planes.  The  common  line  of  inter- 
section is  the  "base"  of  the  sheaf  of  planes. 

A  plane-field  TT  and  a  bundle  of  rays  0'  are  said  to  be  collinear  with 
each  other  if  to  every  point  P  of  TT  there  corresponds  a  ray  p'  of  0',  and 
to  every  straight  line  I  of  TT  that  goes  through  P  there  corresponds  a  plane 
X'  of  the  bundle  0'  that  contains  the  straight  line  p' . 

And,  again: 

Two  bundles  of  rays  0  and  0'  are  said  to  be  collinear  with  each  other, 
if  to  each  ray  p  of  0  there  corresponds  a  ray  pf  of  0',  and  to  each  plane 
X  of  0  that  contains  p  there  corresponds  a  plane  X'  of  0'  that  contains  p'. 

158.     Protective  Relation  of  Two  Collinear  Plane-Fields. 

Two  collinear  plane-fields  r  and  TT'  are  also  called  "protective" ',  be- 
cause to  each  harmonic  range  of  four  points  of  TT  there  corresponds  a 
harmonic  range  of  four  points  of  IT'. 

Thus,  if  P,  Q,  R,  S  (Fig.  82)  are  a  harmonic  range  of  four  points 
of  the  plane-field  TT,  and  if  P',  Qf,  R',  S'  are  the  four  corresponding 
points  of  the  collinear  plane-field  TT',  in  the  first  place,  since  the  points 
P,  Q,  R,  S  all  lie  upon  a  straight  line  s,  the  points  P',  Q',  R',  S'  must 
all  likewise  lie  upon  a  straight  line  sf  which  is  conjugate  to  s.  Let 


§  159.]  The  Geometrical  Theory  of  Optical  Imagery.  203 

A  BCD  be  any  quadrangle  of  the  plane-field  TT,  such  that  the  two 
opposite  sides  A  B  and  CD  intersect  in  the  point  P,  and  the  other  two 
opposite  sides  AD  and  BC  intersect  in  the  point  Q,  while  the  two 
diagonals  BD  and  A  C  go  through  the  points  R  and  5,  respectively. 
To  this  quadrangle  of  TT  there  will  correspond  a  certain  quadrangle 
A'B'C'D'  of  TT',  such  that  the  two  opposite  sides  A' Bf  and  C'D'  inter- 
sect in  the  point  P',  the  other  two  opposite  sides  A'D'  and  B' C 


FIG.  82 
PROJECTIVE  RELATION  OF  Two  COLLINEAR  PLANE-FIELDS. 

intersect  in  the  point  Q',  and  the  two  diagonals  B'D'  and  A' C'  go 
through  the  points  Rr  and  5",  respectively.  Accordingly,  the  points 
P',  <2',  R',  S'  are  also  a  harmonic  range  of  points;  and  this  is  the 
condition  that  the  two  plane-fields  TT  and  TT'  shall  be  projectile. 

By  a  similar  method  we  can  show  also  that  two  collinear  bundles 
of  rays  or  a  bundle  of  rays  and  a  plane-field  in  collinear  relation  are 
projective  to  each  other. 

159.  The  so-called  "Fluent"  Points  of  Conjugate  Rays.  Let  5 
and  s'  denote  two  conjugate  rays  of  the  collinear  plane-fields  x  and 
TT'.  Since,  as  has  just  been  shown,  the  point-ranges  s,  5'  are  projective, 
it  follows  that  the  Double  Ratio  (PQRS)  of  any  four  points  P,  Q,  R,  S 
of  s  is  equal  to  the  Double  Ratio  (P'Q'R'S')  of  the  four  corresponding 
points  P',  Qf,  R',  S'  of  s'.  That  is, 

PR    PS  _P^    P'S' 
QR  ''  QS  "  Q'R' ''  Q'S'' 

'  If  we  suppose  that  P,  Q  and  R  are  three  fixed  points  of  5  and  that 
S  is  a  variable  point,  the  Double  Ratio  (PQRS)  will  vary  in  value 
as  the  point  5  moves  along  s;  and  if  the  point  .S  moves  away  to  an 
infinite  distance  until  it  coincides  with  the  infinitely  distant  point1 

1  Every  actual  straight  line  contains  one  (and  only  one)  infinitely  distant  (or  ideal) 
point,  and  all  rays  having  in  common  the  same  infinitely  distant  point  are  parallel. 


204  Geometrical  Optics,  Chapter  VII.  [  §  160. 

I  of  s,  we  shall  have  : 


where  I'  designates  the  point  on  sf  which  corresponds  to  the  infinitely 
distant  point  /  of  s.  Since  P,  Q  and  R  are  three  actual,  or  finite, 
points  of  s,  no  pair  of  which  are  supposed  to  be  coincident,  the  value 
of  the  ratio  PR  :  QR  is  finite;  and  hence  the  point  I'  conjugate  to  the 
ideal  point  I  of  s  is  a  determinate  and,  in  general,  an  actual,  or  finite, 
point  of  s'. 

Similarly,  if  /'  designates  the  infinitely  distant  point  of  s',  we  shall 
have: 

(PQRJ)  =  (P'Q'R'J')  =         ; 


so  that  the  point  J  which  corresponds  to  the  infinitely  distant,  or  ideal, 
point  J'  of  s'  is,  likewise,  a  determinate  and,  in  general,  an  actual,  or 
finite,  point  of  s. 

In  general,  therefore,  the  points  /  and  I',  corresponding  to  the 
infinitely  distant  points  J'  and  J  of  s'  and  s,  respectively,  are  actual, 
or  finite,  points  having  perfectly  determinate  positions  on  5  and  sr, 
respectively.  In  the  German  treatises  the  points  /  and  /'  are  called 
the  "Flucht"  Points  of  the  two  projective  point-ranges  5  and  s'. 

It  will  be  remarked  that  we  are  careful  to  say  that  the  so-called 
"Flucht"  Points  are  "in  general"  actual,  or  finite,  points;  for  in  one 
special  case,  viz.,  when 

-p  73          p/  r>/ 

(PQRJ)  =  (P'Q'RT)  =         =          ,  > 


the  "Flucht"  Point  I'  will  coincide  with  the  infinitely  distant  point 
/'  of  5',  and  the  "Flucht"  Point  J  will,  likewise,  coincide  with  the 
infinitely  distant  point  /  of  s\  and  in  this  particular  case  the  infinitely 
distant  points  I  and  J'  of  the  projective  point-ranges  s  and  s'  will  also  be 
a  pair  of  conjugate  points. 

160.  The  so-called  "Flucht"  Lines  (or  Focal  Lines)  of  Conjugate 
Planes.  In  the  plane-field  TT  consider  now  a  quadrangle  A  B  CD  (Fig. 
83)  such  that  the  two  pairs  of  opposite  sides  form  two  pairs  of  parallel 
straight  lines.  The  two  parallel  sides  AB  and  CD  intersect  in  the 
infinitely  distant  point  P,  and,  similarly,  the  other  two  parallel  sides 
AD  and  BC  intersect  in  the  infinitely  distant  point  Q-  so  that  if  R 
and  5  designate  the  infinitely  distant  points  of  the  two  diagonals  BD 
and  A  C,  respectively,  the  four  points  P,  Q,  R,  S  are  a  harmonic 


160.] 


The  Geometrical  Theory  of  Optical  Imagery. 


205 


range  of  points  of  the  infinitely  distant,  or  ideal,  straight  line  i  of  the 
plane  ir.1 

In  the  collinear  plane-field  TT'  the  ray  A1 Bf  conjugate  to  AB  will 
go  through  the  point  Pf  conjugate  to  the  infinitely  distant  point  P 
of  the  ray  AB\  so  that  the  point  P'  is  therefore  the  "Fluent"  Point 


FIG.  83. 

THE  "  FLUCHT  "  I,INE  i'  OF  THE  PLANE-FIELD  «•'  CORRESPONDING  TO  THE  INFINITELY  DISTANT 
STRAIGHT  lyiNE  OF  THE  COLLINEAR  PLANE-FIELD  it. 

of  the  ray  A'B'.  Obviously,  the  point  P'  is  also  the  "Fluent"  Point 
of  the  ray  C'D'  conjugate  to  CD.  Precisely  in  the  same  way,  the 
point  Q',  conjugate  to  the  infinitely  distant  point  Q  of  the  parallel 
rays  AD  and  BC,  is  the  common  "Fluent"  Point  of  each  of  the  rays 
A'D'  and  B' C'  conjugate  to  the  rays  AD  and  BC,  respectively.  Let 
R'  and  5'  designate  the  positions  of  the  "Fluent"  Points  of  the  rays 
B'D'  (conjugate  to  BD)  and  A' C'  (conjugate  to  AC),  respectively. 

Since  (§  158)  the  points  P',  Q',  R',  S'  are  a  harmonic  range  of  points, 
they  all  lie  on  a  certain  definite  straight  line  V  of  the  plane-field  TT'; 
and  this  straight  line  if ,  which  is  conjugate  to  the  infinitely  distant  straight 
line  i  of  the  plane-field  ir,  is  the  locus  of  the  "Flucht"  Points  of  all  the 
rays  of  the  plane-field  IT'  collinear  with  TT. 

Similarly,  there  is  a  certain  straight  line  j  of  the  plane-field  TT,  conjugate 
to  the  infinitely  distant  straight  line  j'  of  the  collinear  plane- field  TT', 
which  is  the  locus  of  the  "  Flucht"  Points  of  all  the  rays  of  w. 

German  writers  call  these  two  straight  lines  j  and  i'  the  "  Flucht" 

1  All  the  infinitely  distant  points  of  a  plane  are  assumed  to  lie  in  an  infinitely  distant, 
or  ideal,  straight  line.  The  ideal  line  of  a  plane  must  be  a  straight  line,  because  every 
actual  straight  line  of  the  plane  meets  it  in  only  one  point  —  the  infinitely  distant  point 
of  that  line;  whereas  a  curved  line  may  have  in  common  with  a  straight  line  more  than  one 
point.  Just  as  a  pencil  of  parallel  rays  determines  one  infinitely  distant  point  common 
to  all  the  rays,  so  a  sheaf  of  parallel  planes  determines  one  infinitely  distant  straight  line 
common  to  all  the  planes. 


206  Geometrical  Optics,  Chapter  VII.  [  §  162. 

Lines  (or  "Gegenaxen")  of  the  two  proj active  plane-fields.  We  shall 
designate  them  hereafter,  from  the  stand-point  of  Optics,  as  the  Focal 
Lines  of  the  two  conjugate  planes  TT  and  TT'. 

If  two  plane-fields  are  collinear,  then,  in  general  (that  is,  except  in 
one  particular  case  considered  in  §  161  below),  to  the  infinitely  distant, 
or  ideal,  straight  line  of  one  field  there  corresponds  an  actual,  or  finite, 
straight  line  of  the  other  field,  the  so-called  "Focal  Line"  of  that  field. 

To  a  pencil  of  parallel  rays  in  one  plane-field  there  corresponds 
therefore  a  pencil  of  rays  in  the  other  field  which  all  intersect  in  a 
point  situated  on  the  Focal  Line  of  that  field;  or,  as  we  might  say, 
the  Focal  Line  of  one  plane-field  is  the  locus  of  the  bases  of  pencils  of 
rays  which  are  conjugate  to  pencils  of  parallel  rays  of  the  collinear 
plane-fields 

161.  Affinity  of  Two  Plane-Fields.     The  exceptional  case  men- 
tioned above  cannot  be  passed  over  without  some  explanation.     The 
points  Pf,   Qf,  Rf,  S'  of  TT',  corresponding  to  the  infinitely  distant 
points  P,  Q,  R,  S  of  TT  are,  in  general  (as  was  stated),  actual,  or  finite, 
points,  and  determine,  therefore,  an  actual,  or  finite,  straight  line  i' '; 
except  in  the  one  particular  case  when  the  quadrangle  A' B' C'D',  as 
well  as  the  quadrangle  A  B  CD,  has  each  pair  of  its  opposite  sides 
parallel.     In  this  special  case  the  points  P' ,  Q',  R' ,  S'  will  be  ranged 
along  the  infinitely  distant  straight  line  f  of  the  plane-field  TT',  and 
the  Focal  Line  i'  will  therefore  coincide  with  the  infinitely  distant 
straight  line  j'. 

This  special  case,  in  which  the  two  Focal  Lines  j  and  if  are  also  the 
infinitely  distant  straight  lines  i  and  j'  of  the  collinear  plane-fields  TT 
and  TT',  respectively,  is  the  so-called  case  of  "Affinity"  of  the  two  plane- 
fields. 

This  extremely  important  special  case  will  be  met  with  again. 
Here  we  merely  call  attention  to  it. 

ART.  45.     TWO  COLLINEAR  SPACE-SYSTEMS. 

162.  Two  Space-Systems  1,  and  S'  are  said  to  be  collinear  with  each 
other  if  to  every  point  P  of  2  there  corresponds  one  (and  only  one}  point 
P'  of  S',  and  to  every  straight  line  p  of  2,  which  goes  through  P,  there 
corresponds  one  straight  line  pf  of  S'  which  goes  through  P'. 

It  is  not  necessary  to  think  of  S  and  S'  as  two  separate  and  distinct 
regions  of  space;  they  are  to  be  regarded  rather  as  completely  inter- 
penetrating one  another,  so  that  any  point,  ray  or  plane  in  space  may 
be  considered,  according  to  the  point  of  view,  one  time  as  belonging 
to  the  system  S  and  another  time  as  belonging  to  the  system  S'.  In 


§  163.]  The  Geometrical  Theory  of  Optical  Imagery.  207 

fact,  when  we  say  that  two  space-systems  are  collinear  with  each 
other,  we  mean  that  the  whole  of  space  is  in  collinear  relation  with 
itself;  or  in  the  language  of  the  modern  geometry,  the  whole  of  space 
is  the  common  "base"  of  the  two  space-systems  2,  2'. 

In  the  geometrical  theory  of  Optics  the  two  space-systems  2,  2' 
are  distinguished  as  the  Object-Space  and  the  Image-Space,  respect- 
ively; and  the  points,  rays  and  planes  of  space,  according  as  they  are 
regarded  as  belonging  to  the  one  or  the  other  of  these  two  space- 
systems,  are  called  Object-Points,  Object-Rays  and  Object-Planes  or 
Image-Points,  Image-Rays  and  Image-Planes.  Since  the  relation  be- 
tween the  Object-Space  and  the  Image-Space  is  perfectly  reciprocal, 
there  is  no  essential  difference  between  them;  whence  is  deduced  at 
once  the  theorem  known  as  the  Principle  of  the  Reversibility  of  the 
Light-Path  (§  18). 

A  direct  consequence  of  the  unique  point-to-point  and  ray-to-ray 
correspondence  between  Object-Space  and  Image-Space  is  plane-to- 
plane  correspondence;  so  that  to  every  plane  TT  of  the  Object-Space 
there  corresponds  a  definite  plane  IT'  of  the  Image-Space,  and  vice  versa. 
Thus,  using  the  language  of  the  modern  geometry,  we  may  say: 

In  two  Collinear  Space-Systems  to  every  plane-field  there  corresponds 
a  collinear  plane- field;  to  every  bundle  of  rays  or  planes,  a  collinear 
bundle  of  rays  or  planes;  and  to  every  point-range,  a  protective  point- 
range. 

163.  Two  Space-Systems  2  and  2'  may  be  placed  in  collinear  cor- 
respondence with  each  other  by  taking  any  two  bundles  of  rays  A 
and  B  of  2  and  associating  them  with  any  two  bundles  of  rays  A' 
and  B'  of  2'  in  such  fashion  that  the  rays  AB,  A' B'  common  to  the 
two  pairs  of  bundles  are  corresponding  rays,  and  the  sheaf  of  planes 
AB  of  2  corresponds  with  the  sheaf  of  planes  A' B'  of  2'.  For  if 
this  correspondence  is  established,  and  if  P  designates  a  point  of  2, 
the  pair  of  rays  AP,  BP  determine  a  certain  plane  T\  of  the  sheaf  of 
planes  A  B  to  which  corresponds  in  2'  a  plane  17'  of  the  sheaf  of  planes 
A'B',  and  corresponding  to  the  rays  AP,  BP  of  2,  which  intersect 
in  P,  there  will  be  two  rays  A' P' ,  B' P'  of  2',  which  determine  by 
their  intersection  the  point  P'  of  2'  corresponding  to  any  point  P 
of  2.  Moreover,  corresponding  to  any  ray  5  of  2  projected  from  A 
and  B  by  the  planes  As  and  Bs,  respectively,  there  will  be  a  ray  sf 
of  2'  which  is  determined  by  the  intersection  of  the  two  planes  A's' 
and  B's'  corresponding  to  the  planes  As  and  Bs,  respectively.  And, 
finally,  if  IT  denotes  any  plane-field  of  2  whereby  the  two  bundles  of 
rays  A  and  B  are  in  perspective  with  each  other,  with  the  ray  AB 


208  Geometrical  Optics,  Chapter  VII.  [  §  164. 

common  to  the  two  bundles,  the  two  corresponding  bundles  of  rays, 
A'  and  B' ,  being  also  in  perspective  relation  with  each  other,  with  the 
ray  A' B'  in  common,  will,  accordingly,  determine  a  plane-field  TT'  of 
2'  collinear  with  the  plane-field  TT  of  2.  Therefore,  the  two  Space- 
Systems  2  and  2'  are  placed  in  complete  collinear  correspondence. 

From  this  we  derive  immediately  the  following  rule: 

//  we  take  any  five  points  of  one  Space-System,  no  four  of  which  lie 
in  one  plane,  and  associate  them  as  corresponding  with  five  such  points 
of  the  other  Space- System,  the  two  Space-Systems  will  be  completely  col- 
linear to  each  other. 

Thus,  suppose  we  take  five  points  A,  B,  C,  D,  E  of  2,  no  four 
of  which  lie  in  one  plane,  and  associate  them  with  five  such  points 
A',  B',  C',  D',  E'  of  2',  then  the  two  bundles  of  rays  AB,  AC,  AD, 
AE  and  BA,  BC,  BD,  BE  of  2  correspond  to  the  two  bundles  of 
rays.4'£',  A'C',  A'D',  A' E'  and  B'A',  B'C',  B'D',  B'E',  respectively, 
of  2',  and  to  the  sheaf  of  planes  ABC,  ABD,  ABEotZ  corresponds 
the  sheaf  of  planes  A' B'  C,  A' B'D',  A' B' E'  of  2';  and  we  see,  accord- 
ingly, that  the  rule  given  above  is  equivalent  to  the  method  which 
we  gave  first. 

Since  each  point  of  a  space-pentagon  may  have  a  3 -fold  infinitude  of 
positions,  it  is  obvious  that  two  Space-Systems  may  have  a  1 5-fold  in- 
finitude of  collineations. 

164.  The  so-called  "Fluent"  Planes,  or  Focal  Planes,  of  Two  Col- 
linear Space-Systems.  In  two  collinear  Space-Systems  2  and  2'  let 
TT  and  TT'  designate  two  corresponding  plane-fields,  wherein  KLMN 
and  K'L'M'N'  are  two  corresponding  quadrangles.  The  quadrangle 
KLM  N  determines  a  harmonic  range  of  four  points  P,  Q,  R,  S  which 
all  lie  on  a  straight  line  s  of  the  plane-field  TT;  and,  similarly,  the 
quadrangle  K'L'M'N'  determines  also  a  harmonic  range  of  four 
points  P',  Q',  R',  S',  which  are  conjugate  to  P,  Q,  R,  S,  respectively, 
and  which  all  lie  on  a  straight  line  s'  of  TT'  which  is  conjugate  to  s. 
From  a  point  A  of  2,  lying  outside  the  plane-field  TT,  this  field  is  pro- 
jected by  a  bundle  of  rays  or  planes,  and  from  the  corresponding 
point  A'  of  2'  the  plane-field  TT'  will  be  projected  by  a  bundle  of  rays 
or  planes  which  is  projective  with  the  bundle  A ;  so  that,  for  example, 
the  four  rays  A  P,  AQ,  AR,  AS  and  the  four  corresponding  rays  A' Pf, 
A'Q' ,  A' R' ,  A'S'  form  two  harmonic  pencils  of  rays.  The  complete 
quadrangles  KLMN  and  K'L'M'N'  are  projected  from  A  and  A', 
respectively,  by  two  complete  four-edges. 

Now  suppose  that  the  two  pairs  of  opposite  sides  KL,  M  N  and 
LM,  N  K  of  the  quadrangle  KLM  N  are  two  pairs  of  parallel  straight 


§  165.]  The  Geometrical  Theory  of  Optical  Imagery.  209 

lines,  so  that  the  four  points  P,  Q,  R,  S  are  a  harmonic  range  of 
points  all  lying  on  the  infinitely  distant  straight  line  i  of  the  plane- 
field  TT.  In  the  collinear  plane-field  TT'  the  two  pairs  of  opposite  sides 
of  the  quadrangle  K'L'M'N'  will,  in  general,  not  be  pairs  of  parallel 
straight  lines,  so  that  the  harmonic  range  of  points  P',  Q',  R',  S'  con- 
jugate to  the  infinitely  distant  points  P,  Q,  R,  S  will,  in  general, 
determine  a  finite  straight  line  i' ,  the  so-called  "Flucht"  Line,  or 
Focal  Line  (§  160)  of  the  plane-field  TT'  corresponding  to  the  infinitely 
distant  straight  line  i  of  the  plane-field  TT. 

To  the  plane  Ai  parallel  to  the  plane-field  TT  corresponds  the  plane 
A'i'  of  S',  which,  in  general,  will  not  be  parallel  to  the  plane-field  TT'. 
If  now  the  point  A  is  itself  an  infinitely  distant  point  of  the  Space- 
System  S,  the  corresponding  point  A*  will  be  the  common  "Flucht" 
Point  of  all  the  rays  of  the  bundle  conjugate  to  the  bundle  of  parallel 
rays  of  S  whose  direction  is  determined  by  the  infinitely  distant  point 
A\  and,  in  general,  A'  will  be  a  determinate  and  actual,  or  finite, 
point  of  the  Space-System  S'.  In  this  case  the  plane  Ai  will  be  the 
infinitely  distant  plane1  e  of  S,  and  the  corresponding  plane  A'i'  is 
the  so-called  "Flucht"  Plane  ef  of  S'.  It  contains  the  "Flucht"  Lines 
of  all  the  planes  and  the  "Flucht"  Points  of  all  the  rays  of  Z'. 

Similarly,  there  is  a  certain  plane  <p  of  S,  conjugate  to  the  infinitely 
distant  plane  <p'  of  S',  in  which  are  contained  the  "Flucht"  Lines  of 
all  the  planes  and  the  "Flucht"  Points  of  all  the  rays  of  S. 

These  two  planes  <p  and  e'  are  the  so-called  "Flucht"  Planes  of  the 
two  Space-Systems  S  and  S',  respectively.  In  the  geometrical  theory 
of  optical  imagery  they  play  a  very  important  part,  and  are  called  the 
Focal  Planes  of  the  Object-Space  and  Image-Space.  Hence:  •:  . 

//  we  have  two  collinear  Space-Systems  S  and  S',  which,  in  the  language 
of  Geometrical  Optics,  we  shall  call  the  "Object- Space"  and  the  "Image- 
Space" ,  respectively,  then  (except  in  the  so-called  case  of  Telescopic 
Imagery,  referred  to  below)  to  the  infinitely  distant  (or  ideal)  plane  of 
one  system  there  will  correspond  a  finite  (or  actual)  plane,  the  so-called 
"Flucht"  Plane  or  Focal  Plane,  of  the  other  system. 

Thus,  to  a  bundle  of  parallel  rays  in  one  space  there  will  correspond 
a  bundle  of  rays  in  the  other  space  which  all  intersect  in  a  point  of 
the  Focal  Plane  of  that  space. 

165.  Affinity-Relation  between  Object-Space  and  Image-Space. 
In  the  exceptional  case  when  the  quadrangle  K'L'M'  N' ,  as  well  as 

1  The  infinitely  distant  points  and  lines  of  space  are  assumed  to  lie  in  an  infinitely 
distant  or  ideal  surface,  which,  since  it  is  intersected  by  every  actual  straight  line  in  only 
one  point  and  by  every  actual  plane  in  a  straight  line,  must  be  a  plane  surface — the  in- 
finitely distant  plane  of  space. 
15 


210  Geometrical  Optics,  Chapter  VII.  [  §  166. 

the  quadrangle  KLM N,  has  each  pair  of  its  opposite  sides  parallel, 
so  that  the  points  Pf,  Q' ',  R',  Sf,  corresponding  to  the  infinitely  distant 
points  P,  Q,  R,  S  of  the  plane-field  TT,  are  themselves  also  infinitely 
distant  points  lying  on  the  infinitely  distant  straight  line/  of  the  plane- 
field  ?r',  the  plane  A'j'  conjugate  to  the  plane  Ai  is  parallel  to  the  plane 
TT'.  And  if  also  the  two  corresponding  points  A,  A'  are  infinitely  dis- 
tant points  of  the  Space-Systems  2,  S',  respectively,  then  the  two  Focal 
Planes  tp  and  e'  are  also  the  infinitely  distant  planes  e  and  <p'  of  the  Object- 
Space  and  the  Image-Space,  respectively.  This  case,  which  actually 
occurs  in  certain  optical  systems,  is  called  in  geometry  the  case  of 
"Affinity"  of  the  two  Space-Systems.  In  Optics  it  is  the  important 
case  known  as  "Telescopic  Imagery". 

ART.  46.     GEOMETRICAL   CHARACTERISTICS   OF  OBJECT-SPACE   AND 

IMAGE-SPACE. 

166.  Conjugate  Planes.  The  two  Focal  Planes  <p  and  e'  of  the 
Object-Space  and  Image-Space,  respectively,  not  only  from  the  optical 
but  from  the  geometrical  stand-point  as  well,  are  the  most  distinguished 
planes  of  the  Space-Systems  to  which  they  belong.  The  exceptional 
case  of  Telescopic  Imagery,  alluded  to  in  §165,  in  which  the  Focal 
Planes  <p  and  e'  are  themselves  the  infinitely  distant  planes  of  the 
Space-Systems  S  and  S' ',  respectively,  will  be  treated  specially  and 
in  detail  in  a  separate  division  of  this  chapter.  Therefore  entirely 
excluding  this  case  for  the  present,  and  assuming  that  the  Focal  Planes 
<f>  and  ef  are  finite,  or  actual,  planes,  we  proceed  to  enumerate  the  most 
striking  general  characteristics  of  the  collinear  correspondence  of  two 
Space-Systems  S  and  S'. 

1.  In  general,  to  a  sheaf  of  parallel  planes  of  one  of  the  two  Space- 
Systems  there  will  correspond  a  sheaf  of  non-parallel  planes  of  the  other 
Space-System. 

The  axis  of  the  sheaf  of  parallel  planes  is  an  infinitely  distant  straight 
line  of  the  Space-System  S  to  which  this  sheaf  is  supposed  to  belong; 
and  the  axis  of  the  conjugate  sheaf  of  planes  will  be,  therefore,  a 
straight  line  lying  in  the  Focal  Plane  of  the  other  Space-System  S'. 
Generally  speaking,  this  straight  line  will  be  a  finite,  or  actual,  line  of 
S',  and  such  a  line  can  only  be  the  base  of  a  sheaf  of  non-parallel 
planes. 

However,  there  is  one  very  important  exception  to  the  above  state- 
ment, viz. : 

2.  The  two  sheaves  of  parallel  planes  to  which  the  Focal  Planes  them- 
selves belong  are  conjugate  sheaves. 


§  167.]  The  Geometrical  Theory  of  Optical  Imagery  211 

The  infinitely  distant  straight  lines  of  the  Focal  Planes  <p  and  e' 
are  the  bases  or  axes  of  these  two  sheaves  of  parallel  planes,  and  since 
these  infinitely  distant  straight  lines  are  the  lines  of  intersection  of 
the  Focal  Planes  with  the  infinitely  distant  planes  of  their  respective 
Space-Systems,  they  are  a  pair  of  infinitely  distant  conjugate  straight 
lines,  in  fact  (the  case  of  Telescopic  Imagery  being  excluded)  the  only 
such  pair  of  conjugate  lines.  Hence,  the  two  sheaves  of  parallel 
planes  which  have  these  two  infinitely  distant  conjugate  straight  lines 
'as  axes  are  conjugate  sheaves  of  planes. 

Two  conjugate  planes  a  and  a'  which  are  parallel  to  the  Focal  Planes 
<p  and  ef,  respectively,  are  in  the  relation  of  "Affinity"  to  each  other, 
because  their  infinitely  distant  straight  lines  are  a  pair  of  conjugate 
straight  lines.  Since,  therefore,  to  each  infinitely  distant  point  of 
one  such  plane  there  corresponds  also  an  infinitely  distant  point  of 
the  plane  in  "affinity"  with  it,  it  follows  that: 

Parallel  straight  lines  of  the  plane  a  correspond  to  parallel  straight 
lines  of  the  plane  cr';  so  that  a  parallelogram  in  the  Object-Plane  <r 
will  be  "imaged"  by  a  parallelogram  in  the  Image-Plane  a'. 

Moreover : 

Any  range  of  points  r  of  the  Object-Space,  which  is  parallel  to  the 
Focal  Plane  <p,  will  be  "imaged"  in  a  "protectively  similar"1  range  of 
points  r'  of  the  Image-Space,  which  is  likewise  parallel  to  the  Focal  Plane  e'. 

167.  The  Focal  Points  and  the  Principal  Axes  of  the  Object-Space 
and  the  Image-Space. 

3.  To  a  bundle  of  parallel  rays  in  the  Object-Space  will  correspond, 
in  general,  a  bundle  of  non-parallel  rays  in  the  Image-Space,  the  vertex 
of  which  lies  in  the  Focal  Plane  of  that  space;  and  vice  versa. 

The  particular  point  of  the  Focal  Plane  which  will  be  the  vertex 
of  the  bundle  of  non-parallel  rays  will  depend  on  the  direction  of  the 
bundle  of  parallel  rays.  If,  for  example,  the  bundle  of  parallel  rays 
in  one  space  meets  the  Focal  Plane  of  that  space  at  right  angles,  the 
vertex  of  the  corresponding  bundle  of  rays  in  the  other  space  will 
determine  a  certain  definite  point  in  the  Focal  Plane  of  that  space,  viz., 
the  so-called  Focal  Point  of  that  space.  The  Focal  Point  of  the  Object- 
Space,  designated  by  F,  is  the  vertex  of  the  bundle  of  object-rays  to 
which  corresponds  a  bundle  of  parallel  image-rays  which  cross  the 

1  The  peculiarity  of  "  protectively  similar  "  ranges  of  points  is  that  the  lengths  of  cor- 
responding segments  of  them  are  in  a  constant  ratio  to  each  other.  Thus,  for  example, 
if  r,  r'  are  two  projective  ranges  of  points  whose  infinitely  distant  points  W,  W'  correspond 
to  each  other,  and  if  A,  A'\  B,  B';  C,  C'  are  any  three  pairs  of  conjugate  finite  points  of 
r,  r',  then,  since  (ABCW)  =  (A'B'C'W'),  we  have  immediately  : 

AC:  BC  =  A'C':  B'C',     or     A'C' :  AC  =  B'C' :  BC. 


212  Geometrical  Optics,  Chapter  VII.  [  §  168. 

Focal  Plane  of  the  Image-Space  at  right  angles;  and,  similarly,  the 
Focal  Point  of  the  Image-Space,  designated  by  E' ',  is  the  vertex  of  the 
bundle  of  image-rays  to  which  corresponds  a  bundle  of  parallel  object- 
rays  which  cross  the  Focal  Plane  of  the  Object-Space  at  right  angles. 

The  two  straight  lines  drawn  through  the  Focal  Points  F  and  Ef 
perpendicular  to  the  Focal  Planes  <p  and  e'  are  called  the  Principal 
Axes  of  the  Object-Space  and  the  Image-Space,  respectively.  This 
pair  of  straight  lines  will  be  designated  as  the  axes  of  x  and  x' .  Since 
the  ray  xr  passes  through  the  Focal  Point  £'  (which  corresponds  to 
the  infinitely  distant  point  E  of  x)  and  also  through  the  infinitely 
distant  point  F'  (which  corresponds  to  the  Focal  Point  F  likewise 
situated  on  x),  it  follows  that  the  Principal  Axes  x  and  x'  are  a  pair 
of  conjugate  straight  lines  and,  in  fact,  this  is  the  only  pair  of  conjugate 
rays  which  are  at  right  angles  to  the  Focal  Planes. 

168.  Axes  of  Co-ordinates.  An  immediate  consequence  of  the  fact 
that  x  and  x'  are  a  pair  of  conjugate  rays  is  the  following: 

To  the  sheaf  of  planes  in  the  Object-Space  which  has  for  its  axis  the 
x-axis  corresponds  the  sheaf  of  planes  in  the  Image-Space  which  has  for 
its  axis  the  xf-axis. 

Of  these  two  projective  sheaves  of  so-called  " Meridian  Planes", 
there  is,  according  to  an  elementary  law  of  projective  geometry,  one 
pair  of  Meridian  Object-Planes  at  right  angles  to  each  other  to  which 
corresponds  a  pair  of  Meridian  Image-Planes  which  are  also  at  right 
angles  to  each  other.  In  each  space  this  particular  pair  of  Meridian 
Planes  at  right  angles  to  each  other,  together  with  a  third  plane  per- 
pendicular to  the  Principal  Axis,  and,  therefore,  perpendicular  to  each 
of  the  two  Meridian  Planes,  will  determine  by  their  intersections  a 
set  of  three  mutually  perpendicular  straight  lines.  Hereafter,  when 
we  come  to  derive  the  Image-Equations,  we  shall  find  it  convenient 
to  select  these  two  sets  of  straight  lines  as  the  axes  of  two  systems  of 
rectangular  co-ordinates,  one  in  the  Object-Space  and  the  other  in 
the  Image-Space.  One  of  these  straight  lines  is,  of  course,  the  Prin- 
cipal Axis  x  or  x'  of  the  Space-System.  But,  whereas  x,  x'  will  always 
be  a  pair  of  conjugate  straight  lines,  the  other  two  pairs  of  straight 
lines,  designated  after  the  manner  of  Analytic  Geometry,  as  the  y- 
axis  and  z-axis  in  the  Object-Space  and  the  y'-axis  and  z'-axis  in  the 
Image-Space,  may,  or  may  not,  be  pairs  of  conjugate  straight  lines. 
This  will  depend  on  whether  the  yz-plane  and  the  y'z'-plane  are  a 
pair  of  conjugate  planes.1 

1  Strict  consistency  in  the  matter  of  notation,  which  is  eminently  desirable,  especially 
in  Geometrical  Optics,  cannot,  however,  always  be  observed  without  sacrificing  something 


§  169.]  The  Geometrical  Theory  of  Optical  Imagery.  213 

ART.  47.     METRIC  RELATIONS. 

169.  Relation  between  Conjugate  Abscissae.  Let  s,  sf  be  a  pair 
of  conjugate  rays  of  the  two  collinear  Space-Systems  S  and  2',  and 
let  /  and  /'  designate  the  points  where  these  rays  cross  the  Focal 
Planes  <p  and  e',  respectively.  Moreover,  let  /  and  /'  designate  the 
infinitely  distant  points  of  s  and  s'  conjugate  to  /'  and  J,  respectively. 
Finally,  if  P,  Pr  and  Q,  Q'  are  two  other  pairs  of  conjugate  points 
of  s  and  s',  we  shall  have  : 

(PQJI)  =  (P'Q'JT), 
or 

IPT  PT 


_ 
QJQI~  Q'J'QfI'' 

which,  since  /  and  J'  are  the  ideal  points  of  5  and  5',  reduces  to  the 
following: 

PJ      QT 

QJ  ~  PT  ' 
This  equation  may  be  written  : 

JP-I'P'  =  JQ  -I'Q'  =  a  constant. 

Stated  in  words  this  Characteristic  Metric  Relation  of  Optical  Imagery 
may  be  expressed  as  follows: 

The  product  of  the  "abscissa"  of  two  conjugate  points,  P  and  P', 
with  respect  to  the  so-called  "  Flucht"  Points,  J  and  I',  of  two  conjugate 
rays  s  and  s'  which  go  through  P  and  Pf,  respectively,  is  constant. 

In  this  statement  the  term  "abscissa"  is  employed  (for  lack  of  a 
better  word)  to  describe  the  position  of  a  point  on  a  ray  with  respect 
to  the  "Flucht"  Point  of  the  ray  as  origin.  Thus,  for  example,  the 
"abscissa"  of  the  point  P  of  the  ray  s  is  JP,  which  means  the  segment 
of  the  ray  included  between  J  and  P,  and  reckoned  from  J  to  P,  that 
is,  reckoned  always  in  the  sense  indicated  by  the  order  in  which  the 
letters  are  written.  (See  Appendix,  Art.  4.) 

The  product  of  the  "abscissae"  of  pairs  of  conjugate  points  of  any 
one  pair  of  conjugate  rays  s,  sf  is  constant,  but  the  magnitude  of 

of  greater  importance.  Thus,  according  to  the  system  of  notation  employed  in  this  chapter 
and  very  generally  throughout  this  book,  the  designation  "  yV-plane  "  would  naturally 
imply  a  plane  in  the  Image-Space  conjugate  to  the  yz-plane  in  the  Object-Space.  But 
even  when  these  two  co-ordinate  planes  are  not  conjugate,  we  shall  continue  to  designate 
the  plane  in  the  Image-Space  as  the  yV-plane  rather  than  complicate  and,  perhaps,  con- 
fuse things  by  introducing  a  pair  of  entirely  new  letters.  As  a  matter  of  fact,  except  in 
the  important  case  when  these  planes  yz,  y'z'  are  the  two  Focal  Planes  <j>,  E',  they  are  gen- 
erally a  pair  of  conjugate  planes. 


214  Geometrical  Optics,  Chapter  VII.  [  §  170. 

this  constant,  will  in  general,  be  different  for  different  pairs  of  con- 
jugate rays.  In  particular,  if  the  two  conjugate  rays  are  the  Prin- 
cipal Axes  x,  x'  themselves,  we  shall  have  for  a  pair  of  conjugate  axial 
points  M,  M' : 

FM-  E'M'  =  a  constant; 
so  that,  if  we  put 

x  =  FM,    x'  =  E'M', 

we  obtain  the  so-called  Abscissa- Equation: 

xxf  =  a,  (109) 

where  a  denotes  the  value  of  the  constant  for  the  conjugate  rays  x,  x'. 
170.  The  Lateral  Magnifications.  In  the  special  case  when  the 
object-ray  r  was  parallel  to  the  Focal  Plane  <p  of  the  Object-Space, 
we  saw  (§  1 66)  that  the  corresponding  image-ray  r'  was  likewise  par- 
allel to  the  Focal  Plane  e'  of  the  Image-Space;  so  that,  because  the 
"Fluent"  Points  of  these  rays  coincide  with  their  infinitely  distant 
points,  the  so-called  "Abscissa"  Relation  obtained  in  §  169  is  of  no 
value  when  applied  to  such  a  pair  of  conjugate  rays.  But  the  fact, 
that  to  a  range  of  Object-Points  r  which  is  parallel  to  the  Focal  Plane 
<p  corresponds  a  "projectively  similar"  range  of  Image-Points  r'  which 
is  parallel  to  the  Focal  Plane  e',  leads  also  to  a  very  important  metrical 
relation  concerning  the  rays  r,  r' \  so  that  if  A,  A' \  B,  Bf  \  C,  C'  are 
any  three  pairs  of  conjugate  points  of  r,  r',  we  shall  have: 

A'C      B'C 
AC  =:   BC  * 

This  is  called  the  Magnification- Ratio  for  the  two  conjugate  rays  r,  r' . 

Let  cr,  a'  be  two  conjugate  planes  parallel  to  the  Focal  Planes  <p 
and  e'  and  containing  the  pair  of  conjugate  rays  r,  r' ,  respectively. 
The  plane-fields  cr,  <r',  as  has  been  explained  (§  166),  are  in  the  relation 
of  "affinity"  to  each  other;  so  that  to  a  pencil  of  parallel  rays  of  cr 
corresponds  a  pencil  of  parallel  rays  of  cr';  and,  hence,  for  all  point- 
ranges  of  cr  which  are  parallel  to  r  the  Magnification- Ratio  has  the  same 
value. 

In  §  1 68  it  was  explained  that  there  was  one  pair  of  Meridian  Planes 
in  the  Object-Space  at  right  angles  to  each  other  to  which  corre- 
sponded in  the  Image-Space  a  pair  of  Meridian  Planes  also  at  right 
angles  to  each  other.  The  intersection  of  the  plane  a  with  this  pair 
of  Meridian  Planes  in  the  Object-Space  will  determine  a  pair  of  straight 
lines  of  cr  at  right  angles  to  each  other  which  are  parallel  to  the  co- 


170.] 


The  Geometrical  Theory  of  Optical  Imagery. 


215 


ordinate  axes  y  and  z;  and,  similarly,  there  will  be  determined  in  the 
Image-Space  in  the  same  way  a  pair  of  perpendicular  straight  lines 
of  a'  conjugate  to  those  of  a  which  are  parallel  to  the  co-ordinate 
axes  y'  and  z'  of  the  Image-Space.  If,  therefore,  y,  z  and  y',  z' 
denote  the  co-ordinates  of  a  pair  of  conjugate  points  of  a  and  0-',  so 
that  y,  y'  and  z,  z'  in  this  sense  are  used  to  denote  the  lengths  of  cor- 
responding segments  of  point-ranges  of  cr,  cr'  which  are  parallel  to  the 
axes  y,  y'  and  the  axes  z,  z',  respectively,  the  Magnification-Ratios 
for  rays  of  a  and  a'  parallel  to  these  axes  will  be  yr  /y  and  z'/z. 

These  ratios  y'  \y  and  z'/z  are  called  the  Lateral  Magnifications  for 
the  pair  of  conjugate  planes  <r  and  a' . 

For  a  given  Object-Plane  cr  parallel  to  the  Focal  Plane  <p,  the  two 
Lateral  Magnifications  have  perfectly  definite  values.  Thus,  for  ex- 
ample, the  value  of  y'fy  for  the  plane  <j  may  be  denoted  by  the  symbol 
Y,  so  that 

Y-y- 

~  y' 

which  states  that  the  value  of  Y  is  independent  of  the  actual  magni- 
tudes of  y  and  y'. 

As  origins  of  the  two  systems  of  rectangular  co-ordinates  of  the 
Object-Space  S  and  the  Image-Space  S'  let  us  select  the  two  Focal 


y 


r 
i> 


M, 


N 


FIG.  84. 


COLLINEATION  OF  TWO  SPACE-SYSTEMS  2,2':  showing  how  the  lateral  Magnification  of  Conju- 
gate Planes  parallel  to  the  Focal  Planes  depends  on  the  distances  from  the  Focal  Planes.  The  ranges 
p,  /,  as  drawn  in  this  figure,  are  "  oppositely  projective  "  —  a  case  that  does  not  actually  occur  in 
optical  imagery ;  but  that  fact  is  immaterial  so  far  as  the  question  under  consideration  is  concerned. 

Points  F  and  Er  (Fig.  84),  respectively.  The  Principal  Axes  will  be 
the  axes  of  x  and  x' .  Let  it  be  observed  that  the  Focal  Planes  which 
are  the  planes  yz  and  y'z'  are  not  conjugate  planes,  as  the  notation 
would  imply. 


216  Geometrical  Optics,  Chapter  VII.  [  §  171. 

Take  any  point  P  of  the  Object-Space,  whose  co-ordinates  are 
x  =  FM,  z  =  MN,  y  =  NP,  and  through  P  draw  the  ray  p  parallel  to 
the  tf-axis.  Let  P'  designate  the  point  in  the  Image-Space  conjugate 
to  P,  and  let  x'  =  E'M1,  z'  =  M'N',  y'  =  N' P'  denote  the  co-ordi- 
nates of  P'.  Corresponding  to  the  object-ray  p  going  through  the 
Object-Point  P,  we  shall  have  an  image-ray  p'  which  connects  the 
Image-Point  P'  with  the  Focal  Point  Ef.  The  pair  of  conjugate 
planes  perpendicular  to  the  Principal  Axes  x,  x'  at  the  points  M,  M' 
will  be  designated  by  cr,  a',  and  the  value  of  the  Lateral  Magnification 
for  this  pair  of  planes  and  for  rays  which  are  parallel  to  y  and  y'  will 
be  denoted  by  Y\  so  that  Y  =  y'fy. 

If  the  Object-Point  P  is  supposed  to  move  along  a  straight  line 
parallel  to  the  Focal  Line  y,  it  is  obvious  that  the  Image-Point  P' 
must  also  traverse  a  straight  line  parallel  to  the  Focal  Line  y'  in 
such  fashion  that  the  Lateral  Magnification  y'fy  =  N'P'/NP  =  Y 
shall  be  constant. 

Again,  if  the  Object-Point  P  is  supposed  to  move  along  the  ray  p 
which  is  parallel  to  the  #-axis,  the  Image-Point  P'  will  travel  along 
the  conjugate  ray  p'  which  connects  P'  with  the  Focal  Point  E' ';  so 
that  as  the  ordinate  y  =  NP  remains  constant  as  to  both  magnitude 
and  sign,  its  image  y'  =  N' P'  assumes  all  values  from  —  oo  to  -j-  oo. 

Thus,  it  appears  that  the  Lateral  Magnification  Y  has  different  values 
for  each  pair  of  conjugate  planes  <r,  a'  which  are  parallel  to  the  Focal 
Planes  (p,  e'.  That  is,  the  Lateral  Magnification  Y  is  a  function  of 
the  abscissa  x. 

It  is  obvious  that  the  same  thing  is  true  also  in  regard  to  the  Lateral 
Magnification  z'/z  in  the  direction  perpendicular  to  the  Focal  Line  y. 

171.  The  Image-Equations.  We  proceed,  therefore,  to  ascertain 
in  what  way  the  Lateral  Magnification  Y  depends  on  the  abscissa  x. 
We  shall  continue  to  employ  the  same  symbols  as  in  §  170,  and  shall 
use  the  same  diagram  (Fig.  84).  In  addition  to  the  pair  of  conjugate 
planes  a,  a'  parallel  to  the  Focal  Planes  <p,  e'  and  containing  the  con- 
jugate points  P  (x,  y,  z),  P'(x',  y',  zf),  respectively,  consider  also 
another  pair  of  such  planes  a^  <r\  perpendicular  to  the  Principal  Axes 
x,  x'  at  the  points  Mlt  M(,  respectively.  And  let  7,  Y1  denote  the 
values  of  the  Lateral  Magnification  for  these  two  pairs  of  conjugate 
planes  <r,  </  and  alt  alt  respectively.  Let  the  object-ray  p  parallel 
to  the  x-axis  cross  the  plane  al  at  the  point  Q  whose  co-ordinates  are 
FMl  =  xlt  Ml  Nl  =  zl  =  z,  NiQ  =  y1  =  y.  Similarly,  in  the  Image- 
Space  let  the  ray  p'  conjugate  to  the  object-ray  p  meet  the  plane  <r\  in 
the  point  Q'  whose  co-ordinates  are  E'M (  =  x{,  M^N^  =  z'^N^Q'  =  y\. 


§  171.]  The  Geometrical  Theory  of  Optical  Imagery.  217 

If  the  point  R  of  the  Focal  Plane  <p  is  the  "Fluent"  Point  of  the 
object-ray  p,  then,  by  the  abscissa-relation  of  §  169,  we  have: 

RP-E'P'  =  RQ-E'Q', 


and  since  RP  =  FM  =  x,  RQ  =  FM^  xlt  and  since  from  the  figure 
we  have  also  : 

E'Q'  _  N\Q'      y\ 
E'P'       N'P'~y" 

we  may  write  the  relation  above  as  follows: 


Now  y{  =  Yl-y1  =  Yl-y,  and  Y  =  y'  jy\  accordingly,  we  obtain  finally: 


that  is, 

Y-x  =  a  constant  =  b  (say). 

Thus,  we  see  that  the  Lateral  Magnification  Y  is  inversely  proportional 
to  the  abscissa  x. 

By  precisely  the  same  process  we  should  find  that  the  Lateral 
Magnification  z'  jz  is  also  inversely  proportional  to  the  abscissa  x. 

Accordingly,  we  are  able  now  to  express  the  co-ordinates  x',  y',  z' 
of  any  point  P'  of  the  Image-Space  in  terms  of  the  co-ordinates  x,  y,  z 
of  the  corresponding  point  P  of  the  Object-Space.  Thus,  taking  the 
Focal  Points  F  and  E'  as  the  origins  of  the  two  systems  of  rectangular 
co-ordinates,  and  therefore  using  equation  (109)  together  with  the 
results  which  we  have  just  obtained,  we  can  write  the  Image-  Equations 
as  follows: 

a        .by  cz 

x   =-,     /  =  —  ,     z'  =  —  i  (no) 

X  X  X 

from  which  we  infer  that  the  most  general  case  of  optical  imagery,  as 
defined  by  these  equations,  involves  at  least  three  constants  a,  b  and  c.1 

1  CZAPSKI,  in  his  celebrated  book,  derives  the  Image-Equations  entirely  by  the  methods 
of  Analytic  Geometry.  Taking  as  the  basis  of  his  mathematical  investigation  the  plane- 
to-plane  correspondence  which  is  characteristic  of  the  collinear  relation  of  the  Space- 


218  Geometrical  Optics,  Chapter  VII.  [  §  172. 

III.     COLLINEAR   OPTICAL   SYSTEMS. 
ART.  48.     CHARACTERISTICS  OF  OPTICAL  IMAGERY. 

172.  Signs  of  the  Image-Constants  a,  b  and  c.  Up  to  this  point 
we  have  developed  the  theory  of  Optical  Imagery  from  the  stand- 
point of  pure  geometry,  and  on  this  account,  while  keeping  steadily 
in  view  the  application  to  the  theory  of  optical  instruments,  we  have 
purposely  avoided  introducing  in  this  general  treatment  any  of  the 
physical  properties  of  optical  rays  whereby  the  problem  would  become 

Systems  2  and  2'  and  denoting  the  co-ordinates  of  any  point  P  of  2,  with  respect  to  an 
arbitrary  system  of  rectangular  axes  in  2,  by  x,  y,  z,  and  the  co-ordinates  of  the  conjugate 
point  P',  also  with  respect  to  an  arbitrary  system  of  rectangular  axes  in  2',  by  x',  yr,  z', 
CZAPSKI  shows  that  the  following  equations,  involving  15  independent  constants  (cf  .  end 
of  \  163),  are  the  analytical  expression  of  collinear  correspondence  between  2  and  2': 


,  __ 


,_  azx  +  b.2y  -f 


a4*  +  bty  +  c&  -f  d± 

From  this  system  of  equations  we  may  obtain,  in  general,  also  a  second  system  which 
may  be  written  as  follows: 

-f  «4 


In  each  of  these  two  sets  of  equations  it  will  be  remarked  that  the  right-hand  members 
are  fractions  with  linear  numerators  and  denominators,  and  that  the  denominators  of  the 
fractions  are  identical  for  all  three  equations  in  each  group.  It  is  obvious  that 

a^x  -)-  64y  -f"  c±z  +  ^4=0, 


are  the  equations  of  the  "  Flucht  "  Planes  or  Focal  Planes  <j>,  er  of  the  two  Space-Systems 
2,  2',  respectively. 

Having  thus  obtained  the  equations  above,  CZAPSKI  proceeds  to  show  how  by  a  suitable 
choice  of  axes  of  co-ordinates  the  equations  may  be  reduced  finall}'  to  the  simpler  forms 
given  in  equations  (no),  where,  instead  of  as  many  as  15  independent  constants  in  the 
case  of  arbitrary  systems  of  co-ordinates,  the  number  of  independent  constants  is  only  3. 
See  CZAPSKI:  Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1894),  pages  27-33. 

See  also  E.  WANDERSLEB:  Die  geometrische  Theorie  der  optischen  Abbildung  nach 
E.  ABBE:  Chapter  III  of  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904) 
edited  by  M.  VON  ROHR. 

Also:  JAMES  P.  C.  SOUTH  ALL:  The  Geometrical  Theory  of  Optical  Imagery:  Astrophys. 
Journ.,  xxiv.  (1906),  156-184. 


§  172.] 


The  Geometrical  Theory  of  Optical  Imagery. 


219 


more  or  less  specialized.  But  having  obtained  the  Image-Equations 
(§  171),  we  shall  find  it  convenient  now  to  call  attention  to  the  mani- 
fest singularity  which  distinguishes  optical  rays  from  the  rays  of 
ordinary  geometry.  Along  each  optical  ray  there  is  one  direction, 
viz.,  the  direction  which  the  light  follows,  which  is  the  obvious,  or 
natural,  direction  of  the  ray.  First  of  all,  therefore,  we  may  take 
advantage  of  this  property  by  agreeing  to  define  the  positive  direction 
of  an  optical  ray  as  that  direction  along  the  ray  which  the  light  takes. 

In  the  case,  therefore,  of  two  conjugate  ranges  of  points  s,  s',  there 
are  two  possibilities.  Thus,  for  example,  if  P,  Q,  R,  •  •  •  is  a  series  of 
points  of  s  which  are  traversed  by  the  light  in  the  order  named,  the 
series  of  conjugate  points  P',  (X,  R',-  -  -  lying  on  s'  will  be  traversed 
either  in  the  same  or  in  the  reverse  order.  In  the  former  case  (when 
the  direction  of  the  ray  s'  is  therefore  the  same  as  the  direction  P'Q'), 
we  shall  call  5  and  s'  a  pair  of  directly  protective  ranges  of  points  (Fig. 
85) ;  and  in  the  latter  case  (when  the  direction  of  the  ray  s'  is  opposite 
to  that  of  P'Q'),  we  shall  call  5  and  s'  a  pair  of  oppositely  protective 
ranges  of  points  (Fig.  86).  Obviously,  in  optical  imagery  we  can  have 
only  directly  projective  ranges  of  points,  and,  consequently,  so  far  as 
our  purposes  are  concerned,  we  may  leave  out  of  account  altogether 
oppositely  projective  ranges. 

If,  therefore  P,  P'  are  a  pair  of  conjugate  points  of  the  directly 
projective  ranges  of  points  s,  s'  (Fig.  85),  and  if  /,  /  and  /',  /'  desig- 


FIG.  85. 

DIRECTLY  PROJECTIVE  RANGES  OF  POINTS  ;  SUCH  AS  WE  HAVE  ALWAYS  IN  OPTICAL  IMAGERY. 
O,  O'  are  a  pair  of  conjugate  points,  from  which  the  point-ranges  P,Q,R,---  and  P' ,  &,&,•'•  lying 
on  the  straight  lines  s,  /,  respectively,  are  projected.  The  points  P,  Q,R,  •••  and  P' ',  Of ,  & ,  •  •  •  are 
traversed  by  the  light  in  the  order  in  which  the  points  are  named.  /  and  /'  are  the  "  Flucht" 
Points  and  /  and  /'  the  infinitely  distant  points  of  s,  /,  respectively.  (However,  the  rays  in  the 
diagram  which  are  designated  as  /  and  i'  do  not  here  correspond  to  the  "  Flucht"  I^ines  of  the 
Plane- Fields  *  and  «•' '.) 

nate  the  "Flucht"  Points  and  the  Infinitely  Distant  Points  of  5  and 
s',  respectively,  then,  as  the  point  P  is  supposed  to  travel  along  5 
from  J  to  I  in  the  direction  of  the  ray  s,  P'  will  travel  along  s'  from 


o' 


220  Geometrical  Optics,  Chapter  VII.  [  §  172. 

Jf  to  /'  in  the  positive  direction  of  the  ray  s' ;  so  that,  supposing,  for 
example,  as  is  represented  in  the  figure,  that  the  point  P  lies  on  the 

negative  side  of  the  "Fluent" 
Point  J  (JP  <  o),  the  point 
Pf  will  lie  on  the  positive 
side  of  the  "Fluent"  Point  /' 
(Ir P'  >  o),  and  vice  versa. 

^^  ^x        ^ — ^j-     Hence,  in  the  case  of   two  di- 

FiG.86.  rectly,    or,    as   we   might   say, 

The  range  f.  ff.  # ,  •  •  •  of  points  lying  on  /  is  ' 'optically ' ' ,     prOJCCtive      point- 

oppositely  projective  with  the  range  P,  Q,  R,  •••  ranges,    the    "abscissae"    (§  169) 

lying  on  j  (see  left-hand  side  of  Fig.  85).    This  case  T  T-,     T/  TW      i                i 

cannot  occur  in  optical  imagery.  JP,  /  P'  always  haVC  Opposite 

signs.      Accordingly,    recalling 
the  Abscissa- Relation  derived  in  §  169,  we  may  say: 

In  the  case  of  Optical  Imagery,  the  product  JP  •  I' P'  —  a  constant 
is  always  negative. 

The  value  of  this  constant  for  the  two  projective  point-ranges  lying 
along  the  Principal  Axes  x,  x'  was  denoted  by  a;  hence,  provided  the 
positive  directions  of  the  axes  of  x,  x'  are  defined  as  the  directions  which 
light  pursues  along  these  rays,  the  Image- Constant  a  is  negative  in  all 
cases  of  optical  imagery;  that  is, 

a  <  o. 

In  the  case,  however,  of  a  ray  which  is  parallel  to  the  Focal  Plane, 
the  positive  direction  of  the  ray,  as  denned  above,  is  indeterminate, 
for  the  light  may  be  supposed  to  traverse  such  a  ray  equally  well  in 
either  of  the  two  opposite  directions  of  the  straight  line  to  which  the 
ray  belongs. 

With  regard,  therefore,  to  the  two  systems  of  rectangular  co-ordi- 
nates of  the  Object-Space  and  Image-Space  (§  168),  the  positive  direc- 
tions of  the  Principal  Axes  x,  x'  have  been  clearly  denned ;  but  nothing 
whatever  has  been  done  towards  choosing  the  positive  directions  of 
the  secondary  axes  y,  z  in  the  Object-Space  and  y',  z'  in  the  Image- 
Space.  So  far  as  our  previous  investigation  goes,  the  positive  direc- 
tion of  each  one  of  these  axes  is  entirely  arbitrary;  and,  accordingly, 
the  signs  of  the  two  constants  b  and  c  which  enter  into  the  Image- 
Equations  (no)  may  be  positive  or  negative  and  like  or  unlike,  de- 
pending only  on  the  choice  of  the  positive  directions  of  the  axes  of 
y,  z  and  of  y' ,  z' . 

It  makes  no  difference  which  directions  we  choose  as  the  positive 
directions  of  the  axes  of  y,  z  in  the  Object-Space;  but,  having  chosen 


§  174.]  The  Geometrical  Theory  of  Optical  Imagery.  221 

these,  let  us  contrive  so  that  the  positive  directions  of  the  axes  of 
y',  z'  in  the  Image-Space  shall  be  thereby  determined.  Accordingly, 
we  have  merely  to  make,  for  example,  the  following  agreement: 

The  positive  directions  of  the  axes  of  y  and  y'  are  to  be  chosen  relative 
to  each  other  in  such  manner  that  the  constant  b  shall  be  a  positive  number. 

And  in  the  same  way,  the  positive  directions  of  the  axes  of  z  and  z' 
are  to  be  chosen  with  respect  to  each  other  so  that  the  constant  c  shall  be 
a  positive  number. 

Thus, 

b  >  o,     c  >  o. 

Hence,  assuming  the  positive  directions  of  the  secondary  axes  to  be 
determined  according  to  these  considerations,  it  follows  that  the  Lateral 
Magnifications  y' /y  and  z' /z  always  have  the  same  sign,  viz.,  the  sign 
of  the  abscissa  x. 

The  signs  of  the  three  constants  a,  b  and  c  which  enter  into  the  Image- 
Equations  are  dependent,  therefore,  only  on  the  choice  of  the  positive 
directions  of  the  axes  of  co-ordinates.  If  these  directions  are  denned 
as  above,  then  the  signs  of  these  constants  are  as  follows: 

a  <  o,     b  >  o,     c  >  o. 

173.  So  long  as  we  do  not  assume  any  definite  position-relation 
between  the  Object-Space  and  the  Image-Space,  we  shall  define  the 
positive  directions  of  the  axes  of  co-ordinates  in  this  way,  so  that  a 
is  negative  and  b  and  c  are  positive.     For  the  entirely  general  case 
this  is  the  best  choice  to  make.     But  when  the  optical  system  consists 
of  a  centered  system  of  spherical  refracting  surfaces,  as  is  usually 
the  practical  case,  the  corresponding  axes  of  the  two  systems  of  co-ordinates 
are  parallel,  and  then  it  will  generally  be  more  convenient  to  define  the 
positive  directions  in  such  a  way  that  the  positive  directions  of  cor- 
responding, or  parallel,  axes  will  be  the  same.     If  this  method  is  used, 
the  signs  of  the  Image-Constants  may  be  different,  in  some  cases,  from 
the  signs  which  they  have  above.     It  is  important  to  bear  this  in 
mind,  as  the  student  may  be  puzzled  when  he  finds  that  the  signs  of 
the  Image-Constants  are  sometimes  different  from  the  signs  as  given 
above;  merely  because  the  positive  directions  of  the  axes  of  co-ordi- 
nates have  been  determined  by  different  considerations  (see  §  176). 

174.  Symmetry  around  the  Principal  Axes.     In  the  most  general 
case  of   optical  imagery,  defined  by  equations  (no),  which  involve 
at  least  as  many  as  three  constants  a,  b  and  c,  the  imagery  is  not 
symmetrical  with  respect  to  the  Principal  Axes  of  the  Object-Space 


222  Geometrical  Optics,  Chapter  VII.  [  §  174- 

and  Image-Space;  that  is,  in  general,  the  two  Magnification-Ratios 
y'/y  and  z'/z  have  different  values  corresponding  to  the  same  value 
of  x.  However,  in  most  actual  optical  systems,  in  fact  almost  with- 
out exception,  the  Principal  Axes  are  axes  of  symmetry;  and,  since 
we  are  concerned  primarily  with  the  applications  of  these  laws  to  the 
theory  of  optical  instruments,  it  will  be  assumed  hereafter  that  this 
is  the  case.  Thus,  we  shall  put 

c  =  b\ 

in  which  case  the  I  mage-  Equations  become  : 

a  by  bz  /      \ 

x'  =  -,     y  =  -A     z'  =  —  ;  (in) 

XXX 

so  that  the  character  of  the  imagery  will  be  defined  now  by  the  two 
constants  a  and  b.  The  Principal  Axes  being  axes  of  symmetry, 
every  pair  of  Meridian  Planes  of  the  Object-Space  which  are  at  right 
angles  to  each  other  has  a  corresponding  pair  of  Meridian  Planes  of 
the  Image-Space  also  at  right  angles  to  each  other;  so  that  the  choice 
of  the  axes  of  y  and  z  is  now  indeterminate. 

Moreover,  in  the  case  of  symmetry  with  respect  to  the  Principal 
Axes  of  x,  x',  when  we  have  c  =  b,  the  collinear  plane-fields  o-,  a' 
parallel  to  the  Focal  Planes  <p,  e',  respectively,  are  not  only  in  affinity 
with  each  other  (§  166),  but  they  are  also  similar;  so  that  if  A,  B,  C 
are  three  points  of  a  not  in  the  same  straight  line,  and  -4',  Br,  Cf 
the  three  corresponding  points  of  c/,  then 

A'B'      B  A'C 


AB   ''   BC        AC 

and,  consequently,  corresponding  angles  of  two  similar  plane-fields  are 
equal. 

The  simplest,  and  at  the  same  time  the  most  perfect,  kind  of  optical 
image  would  be  one  which  was  geometrically  exactly  similar  to  the 
object;  so  that  it  would  always  be  possible  to  conceive  the  object 
and  image  oriented  with  respect  to  one  another  in  such  fashion  that 
all  corresponding  lines  were  parallel.  This  case  of  complete  geomet- 
rical similarity  between  an  object  and  its  optical  image  may  not,  in 
general,  be  realized  by  any  optical  apparatus,  although  it  is  possible 
in  special  cases.  But  if  the  object  is  a  plane  figure  lying  in  a  plane 
parallel  to  the  Focal  Plane  of  the  Object-Space,  the  image  will  be  a 
completely  similar  figure  lying  in  a  plane  parallel  to  the  Focal  Plane 
of  the  Image-Space  —  assuming  that  we  have  collinear  correspondence 


§  175.]  The  Geometrical  Theory  of  Optical  Imagery.  223 

between  Object-Space  and  Image-Space,  and  that  the  Principal  Axes 
are  axes  of  symmetry. 

175.  The  Different  Types  of  Optical  Imagery.1  In  collinear 
bundles  of  rays  there  are  always  two  corresponding  rectangular  three- 
edges.  Thus,  at  any  point  0  of  the  Object-Space,  let  OA,  OB,  OC 
be  three  rays  mutually  at  right  angles  to  each  other,  to  which  there 
correspond  three  rays  O'A',  O'B',  0' C'  meeting  at  the  conjugate 
point  Of  of  the  Image-Space,  and  also  mutually  at"  right  angles  to 
each  other.  Let  us  suppose  that  the  three  edges  OA,  OB,  OC  of  the 
octant  O-A  B  C  form  a  canonical  or  right-screw  system  of  axes  (so  that, 
if  a  right-screw  was  turned  in  the  direction  from  OB  to  OC,  the  point 
of  the  screw  would  advance  along  OA).  Two  cases  may  occur,  as 
follows:  (i)  The  system  O'A',  O'B',  0' C'  may  also  be  a  right-screw 
system;  or  (2)  The  system  O'A',  O'B',  0' C'  may  be  a  left-screw 
(or  acanonical)  system  of  axes. 

In  the  first  case,  we  can  see  how  it  might  be  possible,  by  placing 
the  points  0  and  0'  together,  to  fit  one  of  the  octants  into  the  other 
in  such  fashion  that  the  directions  of  the  three  pairs  of  corresponding 
edges  of  the  two  conjugate  octants  OA,  O'A'-,  OB,  O'B';  OC,  0' C' 
agree  with  one  another;  so  that  except  for  the  fact  that  the  pairs  of 
corresponding  points  A,  A'-,  B,  B'-,  C,  C'  will  not,  in  general,  be 
superposed  on  each  other,  we  should  have  "congruence"  of  the  two 
rectangular  corners  O-A  B  C  and  O'-A'B'C'. 

In  the  latter  case,  when  one  system  is  canonical  and  the  other 
acanonical,  no  such  "congruence"  would  be  possible.  Thus,  for 
example,  if  in  this  case  we  place  the  points  O  and  0'  in  coincidence 
with  each  other,  and  if  we  orient  the  two  octants  relative  to  each  other 
so  that  the  directions  of  two  pairs  of  conjugate  edges,  say,  OB,  O'B' 
and  O  C,  Of  C'  are  the  same,  the  directions  of  the  third  pair  of  edges 
OA,  O'A'  will  be  exactly  opposite  to  each  other.  Instead,  therefore, 
of  a  so-called  "congruence"  of  the  two  conjugate  octants  0-ABC 
and  O'-A'B' C',  such  as  was  possible  in  the  first  case,  we  shall  have 
here  a  certain  "symmetry"  of  the  two  octants;  although  here  again 
we  are  employing  a  term  in  a  sense  somewhat  different  from  the  pre- 
cise meaning  attached  to  it  in  geometry.  Strictly  speaking,  both 
"congruence"  and  "symmetry"  involve  the  idea  of  the  equality  of 
corresponding  line-segments,  which  is  by  no  means  necessarily  implied 
in  the  employment  of  these  terms  in  the  present  connection. 

The  following  discussion  is  based  on  the  admirable  treatment  of  this  matter  in  E. 
WANDERSLEB'S  article  on  "  Die  geometrische  Theorie  der  optischen  Abbildung  nach  E. 
ABBE",  which  is  Chapter  III  of  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin. 
1904),  edited  by  Dr.  M.  VON  ROHR.  See  pages  92,  foil. 


224  Geometrical  Optics,  Chapter  VII.  L  §  175. 

By  virtue  of  the  Principle  of  Continuity,  it  is  obvious  that  if  we 
have  "congruence",  or  "symmetry",  between  one  pair  of  conjugate 
octants,  we  shall  have  "congruence",  or  "symmetry",  between  all 
pairs  of  conjugate  octants.  It  is  true  that  possibly  at  the  Focal 
Planes  (which  are  sometimes  called  the  "Discontinuity  Planes")  of 
the  two  Space-Systems,  the  imagery  might  change  from  "congruence" 
to  "symmetry",  or  vice  versa.  That  this  change  does  not  occur  in 
crossing  from  one  side  of  the  Focal  Plane  to  the  other,  we  shall  now 
proceed  to  show. 

Let  us  assume  that  the  two  pairs  of  conjugate  points  0,  0'  and 
A,  A'  are  situated  on  the  Principal  Axes  x,  xf ;  so  that  OB,  0' Bf  and 
OC,  0'  C'  are  parallel  to  the  co-ordinate  axes  y,  yf  and  z,  z',  respectively. 
Moreover,  let  us  assume  also  that  the  points  A,  B,  C  are  all  infinitely 
near  to  0;  and,  consequently,  the  points  A',  B',  C'  will  also  be 
infinitely  near  to  0'.  Hence,  if 

x  =  FO,     x'  =  E'O' 

denote  the  abscissae  of  the  points  0,  0'  with  respect  to  the  Focal  Points 
F,  E',  respectively,  as  origins,  we  shall  have : 

OA  =  dx,        OB  =  dy,        OC  =  dz\ 
O'A'  =  dx',     O'B'  =  dy',     O'C  =  dz'. 

And,  finally,  let  us  suppose  that  the  directions  OA,  OB,  OC  agree 
with  the  positive  directions  of  the  axes  of  x,  y,  z,  respectively  (no 
matter  how  these  directions  may  have  been  defined) ,  so  that  the  magni- 
tudes denoted  here  by  dx,  dy,  dz  are  all  positive.  If  we  write  the 
Image-Equations  (no)  in  the  differential  form,  as  follows: 

dx'  =  —  ~z  dx,     dy'  =  -  dy,    dz'  =  -  dz, 

+  OC  OC  OC 

we  see  that,  as  the  point  O  is  supposed  to  cross  the  Focal  Plane  at 
F,  whereas  the  abscissa  x  changes  its  sign,  the  sign  of  dx'  remains  the 
same;  but  the  signs  of  dy'  and  dz'  both  change  with  change  of  the 
sign  of  x.  Consequently,  the  octant  O'-A'B'  C'  remains  of  the  same 
type,  so  that  if  it  was  "congruent"  (or  "symmetric")  with  the  octant 
O-A  B  C  when  the  point  O  was  on  one  side  of  the  Focal  Plane,  it  will 
remain  "congruent"  (or  "symmetric")  with  it  when  the  point  0  is 
taken  on  the  other  side  of  the  Focal  Plane. 

Accordingly,  from  this  purely  geometrical  standpoint,  and  entirely 
without  reference  to  the  actual  signs  of  the  Image-Constants  a,  b,  c, 


§  175.] 


The  Geometrical  Theory  of  Optical  Imagery. 


225 


it  appears  that  there  are  these  two  essentially  different  types  of  optical 
imagery,  which  may  be  conveniently  distinguished  as  follows: 

1.  Right- Screw  Imagery — the  case  when  the  two  conjugate  octants 
are  ' 'congruent";  in  this  case  the  image  of  a  right-screw  will  be  a 
right-screw,  but,  in  general,  distorted;  and 

2.  Left-Screw  Imagery — the  case  when  the  two  conjugate  octants 
are  "symmetric";  in  this  case  the  image  of  a  right-screw  will  be  a 
left-screw,  although,  in  general,  distorted. 

These  two  types  of  imagery  may  be  exhibited  by  diagrams  (Figs. 
87,  88  and  89)  as  follows: 

In  the  Object-Space  parallel  to  the  x-axis  draw  two  pairs  of  rays, 
viz.,  two  rays  b,  b  (Fig.  87)  in  the  #;y-plane  at  equal  distances  from, 


B, 


/o,       /oa       /o.7       /F       /o,      /of      / 


FIG.  87. 

TYPES  OF  OPTICAL  IMAGERY  :  OBJECT-SPACE.  This  figure  shows  a  series  of  equidistant  axial 
Object-Points  O\,  O2,  etc.  and  two  series  of  equidistant  Object-Points  B\,  £*,  etc.  and  C\,  C2,  etc., 
lying  in  the  planes  xy  and  xz,  respectively,  on  the  straight  lines  b  and  c  parallel  to  the  Principal 
Axis  (x)  of  the  Object-Space. 

and  on  opposite  sides  of,  the  x-axis;  and,  similarly,  two  rays  c,  c 
drawn  in  the  same  way  in  the  #z-plane.  In  the  Image-Space  (Figs. 
88  and  89),  corresponding  to  the  two  pairs  of  object-rays  b,  b  and  c,  c, 
we  have  two  pairs  of  image-rays  &',  b'  and  c' ,  c'  all  passing  through  the 
Focal  Point  £'.  The  pair  of  rays  &',  b'  will  lie  in  the  x'y'-plane, 
and  the  pair  of  rays  c',  c'  will  lie  in  the  #V-plane,  and  the  #'-axis 
will  bisect  the  angles  at  Ef  between  each  of  the  pairs  of  rays  b',  b' 
and  c',  cr.  In  Fig.  87  Olt  02,  etc.  represent  a  series  of  equidistant 
Object-Points  ranged  along  the  #-axis.  Through  each  one  of  these 
points  draw  a  pair  of  lines  parallel  to  the  axes  of  y  and  z,  and  con- 
sider the  segments  of  these  lines  comprised  between  £>,  b  and  c,  c. 
The  image  of  one  of  these  rectangular  crosses  made  by  such  a  pair  of 
line-segments  will  be  a  rectangular  cross  with  its  arms  parallel  to  the 

16 


226 


Geometrical  Optics,  Chapter  VII. 


[  §  175. 


axes  of  y  and  z'\  the  end-points  of  these  arms  lying  in  the  pairs  of 
rays  b',  b'  and  cf,  c',  as  shown  in  Figs.  88  and  89.  The  points  0{,  0'2, 
etc.,  corresponding  to  the  axial  Object- Points  Olt  02,  etc.  (Fig.  87), 


FIG.  88. 

TYPES  OF  OPTICAL  IMAGERY  :  IMAGE-SPACE.    This  figure  is  to  be  taken  in  connection  with  Fig. 
87.    It  shows  the  case  of  Right-Screw  Imagery. 

will  be  ranged  along  the  #'-axis,  and  will  lie  nearer  to  the  Focal  Point 
E'  in  the  same  proportion  as  the  object-points  Olt  02,  etc.  are  farther 
from  the  Focal  Point  F,  and  vice  versa. 

In  Figs.  87  and  88  the  imagery  is  right-screw  imagery;  whereas  in 
Figs.  87  and  89  the  imagery  is  left-screw  imagery.  The  directions  of 
the  line-segments  are  shown  by  the  arrow-heads.  In  these  diagrams 


PIG.  89. 

TYPES  OF  OPTICAL  IMAGERY  :  IMAGE-SPACE.    This  figure  is  to  be  taken  in  connection  with  Fig. 
87.    It  shows  the  case  of  lyeft-Screw  Imagery. 

the  positive  directions  of  the  axes  are  chosen  so  that  the  signs  of  the 
Image-Constants  (§  172)  are  given  by  the  following  relations: 

a  <  o,     b  >  o,     c  >  o; 

and,  consequently,  corresponding  to  positive  values  of  the  co-ordinates 
x,  y,  z,  we  shall  have  x'  negative  and  yf  and  z'  positive;  whereas  if 
y  and  z  are  both  positive,  but  x  negative,  x'  will  be  positive,  and  yr 
and  z'  both  negative. 


§  176.]  The  Geometrical  Theory  of  Optical  Imagery.  227 

One  of  the  most  obvious  and  characteristic  features  of  optical 
imagery  is  the  symmetry  of  the  imagery  with  respect  to  the  two  Focal 
Planes.  Each  of  the  two  space-regions  is  divided  by  its  Focal  Plane 
into  two  equal  halves,  and  to  each  half  of  the  Object-Space  corre- 
sponds one  of  the  two  halves  of  the  Image-Space. 

176.  In  order  not  to  affect  the  generality  of  our  results,  up  to 
this  point  we  have  purposely  nowhere  assumed  any  definite  position- 
relation  between  the  Object-Space  and  the  Image-Space.  As  a  matter 
of  fact,  however,  practically  all  optical  instruments  consist  of  a 
centered  system  of  spherical  refracting  (or  reflecting)  surfaces,  so  that 
the  system  is  perfectly  symmetrical  with  respect  to  the  optical  axis 
(§  135),  or  straight  line  along  which  lie  the  centres  of  the  spherical 
surfaces.  In  such  a  system  the  Principal  Axes  x,  xr  of  the  Object- 
Space  and  the  Image-Space  are  both  coincident  with  the  optical  axis. 
A  ray  lying  in  a  Meridian  Plane  of  the  Object-Space  must  in  its  transit 
through  the  system  continue  to  lie  always  in  this  same  plane  in  space, 
so  that  a  Meridian  Object- Plane  and  its  conjugate  Image-Plane  are 
the  same  plane  in  space.  Thus,  for  example,  the  two  Meridian  Planes 
of  the  system  of  co-ordinates  of  the  Object-Space,  viz.,  the  planes  xy 
and  xz,  are  coincident  with  the  planes  x'y'  and  x'z',  respectively,  of 
the  Image-Space.  Hence,  the  axes  of  y  and  z  in  the  Object-Space  are 
parallel  to  the  axes  of  y'  and  zf,  respectively,  in  the  Image-Space.  This 
being  the  case,  it  is  usually  found  convenient  to  select  the  positive  direc- 
tions of  the  axes  of  x',  y',  z'  so  that  these  directions  shall  be  the  same  as 
the  positive  directions  of  the  axes  x,  y,  z,  respectively.  Thus,  while  we 
shall  always  select  the  positive  direction  of  the  re-axis  as  the  direction 
taken  by  the  incident  light  along  that  line  (§  172),  the  positive  direc- 
tion of  the  jc'-axis  may,  or  may  not,  be  the  direction  pursued  by  the 
light  along  it.  And,  therefore,  the  constant  a  may  in  a  case  of  this 
kind  be  either  positive  or  negative,  depending  on  which  direction  of 
the  x'-axis  is  the  positive  direction  (see  §  173). 

In  an  optical  system  composed  of  a  centered  system  of  spherical 
surfaces,  it  is  important  to  emphasize  the  fact  that  the  positive  direc- 
tion along  the  optical  axis  is  always  the  direction  of  the  incident  light; 
so  that,  for  example,  if  one  of  the  spherical  surfaces  is  a  reflecting 
surface  whereby  the  original  direction  of  the  light  along  the  optical 
axis  is  reversed,  notwithstanding,  we  must  continue  to  reckon  as 
positive  that  direction  which  was  originally  the  positive  direction; 
and  all  axial  line-segments,  irrespective  of  any  subsequent  change  of 
the  direction  of  the  light,  are  to  be  reckoned  as  positive  or  negative 
according  as  they  have  the  same  direction  as,  or  the  opposite  direction 
to,  the  incident  axial  ray  (see  §§26  and  108). 


228  Geometrical  Optics,  Chapter  VII.  [  §  176. 

So  also  in  regard  to  the  other  Image-Constant  b  =  c:  since  here 
we  do  not,  as  in  the  general  case,  choose  the  positive  directions  of 
the  secondary  axes  of  yf  and  z'  so  that  b  =  c  shall  be  positive,  the 
Image-Constant  b  =  c  may,  therefore,  be  positive  or  negative.  In 
brief,  in  these  special  circumstances,  the  two  systems  of  axes  are 
chosen  with  respect  to  each  other  so  that  a  mere  displacement  along 
the  optical  axis  of  the  origin  of  co-ordinates  of  the  Image-Space  is 
all  that  is  needed  in  order  to  bring  the  axes  of  co-ordinates  of  the 
Image-Space  into  coincidence  with  the  axes  of  co-ordinates  of  the 
Object-Space. 

If  we  write  again  the  Image-Equations  in  their  differential  forms, 
viz.: 

dxf  —  —  -^dx,    dy'  =  -dy,    dz'  =  -dz, 

and  assume  always  that  dx,  dy,  dz  are  positive,  we  may  consider  the 
following  cases: 

I.  As  to  the  sign  of  the  constant  a: 

(1)  If  a  <  o,  then  whatever  may  be  the  sign  of  the  abscissa  x,  the 
sign  of  dx'  must  be  positive.     The  signs  of  dy'  and  dz'  are  always 
either  both  positive  or  both  negative,  depending  on  the  sign  of  x. 
Consequently,  the  two  conjugate  octants  which  have  dx,  dy,  dz  and 
dx',  dy',  dz'  as  corresponding  edges  are  "congruent",  and,  hence,  when 
a  <  o,  we  have  Right-Screw  Imagery  (§  175). 

(2)  When  a  >  o,  the  sign  of  dx'  must  be  negative  for  both  positive 
and  negative  values  of  x;  whereas,  as  before,  the  signs  of  dyf  and  dz' 
are  either  both  positive  or  both  negative,  depending  on  the  sign  of  x; 
so  that  the  two  conjugate  octants  which  have  dx,  dy,  dz  and  dxf,  dy', 
dz'  as  corresponding  edges  are  "symmetric"  (§  175).     Hence,  when 
a  >  o,  we  have  Left-Screw  Imagery. 

II.  As  to  the  sign  of  the  constant  b  —  c: 

(1)  When  b  >  o,  the  signs  of  dy'  and  dz'  are  the  same  as  that  of  x. 
Accordingly,  for  positive  values  of  x,  we  have  erect  images,  and  for 
negative  values  of  x,  we  have  inverted  images.     An  optical  system  of 
this  kind  is  called  a  convergent  system. 

(2)  When  b  <  o,  the  signs  of  dy'  and  dz'  are  opposite  to  that  of  x\ 
so  that  the  positive  half  of  the  Object-Space  is  portrayed  by  inverted 
images,  whereas  the  other  half  (the  negative  half)  is  portrayed  by 
erect  images.     This  case  is,  accordingly,  precisely  opposite  to  the  one 
above,  and  a  system  of  this  kind  is  called  divergent. 

These  results  may  be  summarized  as  follows: 


§  177.]  The  Geometrical  Theory  of  Optical  Imagery.  229 

A  centered  system  of  spherical  refracting  (or  reflecting)  surfaces  is 
convergent  or  divergent  according  as  the  Image- Constant  b  >  or  <  o; 
and  the  Imagery  is  Right-Screw  or  Left-Screw  Imagery  according  as  the 
other  Image-  Constant  a  <  or  >  o. 

ART.  49.     THE    FOCAL   LENGTHS,    MAGNIFICATION-RATIOS,    CARDINAL 

POINTS,  ETC. 

177.  Analytical  Investigation  of  the  Relation  between  a  Pair  of 
Conjugate  Rays.  Let  the  Focal  Point  F  (Fig.  90)  be  the  origin  of 


FIG.  90. 

RELATION  OF  OBJECT-RAY  AND  CONJUGATE  IMAGE-  RAY.  The  figure  shows  only  the  object-ray  ; 
a  similar  diagram,  with  letters  suitably  changed,  may  be  imagined  for  the  image-ray.  PQ  (or  s) 
represents  an  object-ray  which  crosses  the  Focal  Plane  yz  at  the  point  designated  by  R.  FR  =  g, 
this  distance  being  reckoned  positive  or  negative  according  as  R  is  above  or  below  the  Jtr^-plane. 
The  angle  9  may  have  any  value  between  irl2  and  —  irl2;  the  sign  of  this  angle  being  always  the 
same  as  that  of  the  angle  ty  =  ^  FLU,  where  PU\s  the  projection  of  PQ  on 


the  system  of  rectangular  co-ordinates  of  the  Object-Space,  the  Prin- 
cipal Axis  of  the  Object-Space  being  the  x-axis,  and  the  Focal  Plane 
(p  being  the  yz-plane.  Similarly,  in  the  Image-Space  (not  represented 
in  the  figure)  the  Focal  Point  E'  is  the  origin  of  a  system  of  rectangular 
axes,  the  Principal  Axis  of  the  Image-Space  being  the  #'-axis,  and  the 
Focal  Plane  e'  being  the  y'z'-plane.  Consider  a  pair  of  conjugate 
rays,  an  Object-Ray  (s),  which  crosses  the  co-ordinate-planes  xy,  xz, 
yz,  at  the  points  designated  in  the  figure  by  the  letters  P,  Q,  R,  re- 
spectively, and  the  corresponding  Image-  Ray  ($'),  which  crosses  the 
co-ordinate-planes  x'y',  x'z'  ,  y'z'  of  the  Image-Space  at  the  points 
N',  0',  S'j  respectively.  Draw  R  U  perpendicular  to  the  ^-axis  and 
S'V  perpendicular  to  the  y'-axis;  then  the  straight  lines  PU  and 
N'V  meeting  the  #-axis  in  the  point  L  and  the  #'-axis  in  the  point 


230  Geometrical  Optics,  Chapter  VII.  [  §  177. 

Mf,  will  be  the  projections  on  the  ;ry-plane  and  the  #';y'-plane  of  the 
conjugate  rays  s,  s',  respectively. 

The  co-ordinates  of  the  points  R  and  S'  where  the  object-ray  (s) 
and  the  image-ray  (s')  cross  the  Focal  Planes  <p  and  e',  respectively, 
will  be: 

(o,  FU,  UR)     and     (o,  E'V,  V'S')  ; 

and,  hence,  if  /,  m,  n  and  I',  m' ,  n'  are  the  direction-cosines  of  the 
straight  lines  s,  s',  respectively,  the  Cartesian  equations  of  these 
straight  lines  will  be: 

g"  UR 


m  n 

y'  -  E'V'      z'  - 


7'  m'  n' 

respectively. 

Assuming  that  the  imagery  is  symmetrical  with  respect  to  the 
Principal  Axes  x,  xr  ,  so  that  b  =  c  (§  174),  we  can  express  the  relations 
between  the  co-ordinates  x,  y,  z  of  an  Object-Point  and  the  co-ordinates 
x'j  y'j  z'  of  the  conjugate  Image-Point  by  means  of  the  Image-Equa- 
tions (in);  in  consequence  whereof  the  second  pair  of  the  above 
equations  may  be  written  as  follows: 

E'V'         a  m'  V'S'        a  n' 


Comparing  this  pair  of  equations  with  the  first  pair  above,  we  obtain 
immediately  the  following  relations  for  the  co-ordinates  of  the  two 
points  R  and  Sr  in  the  Focal  Planes  yz  and  /z',  respectively: 


j,      V'S'  =  b". 


Let  us  denote  the  focal  distances  of  the  points  R,  S'  where  s,  s'  cross 
the  Focal  Planes  <p,  e'  by  g,  k',  respectively  ;  that  is,  FR  =  g,  E'S'  =  k'  ; 
and,  moreover,  let  B,  0'  denote  the  angles  of  inclination  to  the  axes 
x,  x'  of  the  conjugate  rays  5,  s',  respectively.  Squaring  and  adding 
the  two  equations  in  the  top  line,  and  doing  the  same  for  the  two 


§  177.]  The  Geometrical  Theory  of  Optical  Imagery.  231 

equations  in  the  lower  line,  and  introducing  the  symbols  which  we 
have  just  defined,  at  the  same  time  remarking  that  we  have  also: 

and 

/  =  cos  0,     /'  =  cos  0', 

we  obtain  immediately  the  following  results: 

I 
Thus,  we  find: 

t  - 
g=        b 

In  order  to  avoid  ambiguity  of  signs  in  this  pair  of  equations,  it 
is  necessary  to  define  more  precisely  the  linear  magnitudes  g  and  k' 
and  the  angular  magnitudes  0  and  0'. 

1.  As  to  the  signs  of  the  linear  magnitudes  g  and  k' :   The   focal 
distances  g  and  k'  are  to  be  reckoned  positive  or  negative  according 
as  their  projections  FU  and  E'V  on  the  y-axis  and  y'-axis,  respec- 
tively, are  positive  or  negative.     Thus,  according  as  the  point  U  lies 
on  the  positive  or  negative  half  of  the  ^-axis,  the  sign  of  g  will  be  plus 
or  minus;   and,  according  as  the  point   V  lies  on  the  positive  or 
negative  half  of  the  y~axis,  the  sign  of  k'  will  be  plus  or  minus. 

2.  As  to  the  angular  magnitudes  0  and  0':   If  through  the  point 
P  where  the  object-ray  meets  the  #;y-plane  a  straight  line  is  drawn 
parallel  to  the  #-axis,  in  the  same  direction  as  the  positive  direction 
of  the  #-axis,  the  angle  0  is  the  acute  angle  through  which  this  straight 
line  has  to  be  turned  about  P  in  order  to  bring  it  into  coincidence 
with  the  straight  line  PQ.     The  s;gn  of  this  angle  may  be  positive  or 
negative,  its  value  being  comprised  between  0  =  ir/2  and  0  =  —  7r/2. 
The  sign  of  the  angle  0  can  always  be  ascertained  by  the  following 
rule:  If  $  =  /.FLP  denotes  the  acute  angle  through  which  the  tf-axis 
must  be  revolved  about  the  point  L  in  order  to  make  it  coincide  in 
position  with  the  projection  PL  of  the  object-ray  PQ  on  the  jry-plane, 
and  if  the  sign  of  the  angle  ^  is  determined  by  the  relation 

-  — 
FL ' 

let  us  agree  that  the  signs  of  the  angles  here  denoted  by  0  and  \f/  shall 
always  be  the  same.     Thus,  for  example,  in  the  figure,  as  it  is  drawn, 


232  Geometrical  Optics,  Chapter  VII.  [  §  178. 

both  F  U  and  FL  are  positive,  since  their  directions  are  the  same  as 
the  positive  directions  of  the  axes  of  y  and  x,  respectively;  and  hence 
the  angle  0  in  the  figure  is  negative. 

The  angle  6'  in  the  Image-Space  is  defined  in  an  entirely  similar 
way. 

If  the  pair  of  conjugate  rays  lie  in  a  pair  of  conjugate  Meridian 
Planes,  we  shall  find,  on  investigation,  that  it  will  not  be  necessary 
to  extract  a  square-root,  as  it  was  in  the  general  case  above,  and  that, 
with  the  above  definitions  of  the  magnitudes  denoted  by  g,  k' ,  6,  0', 
the  positive  sign  in  the  two  formulae  is  alone  admissible.  Thus,  the 
ambiguity  disappears,  and  we  must  write : 

g  =  r-tan0',     k'  =  b-tan6.  (112) 

From  these  formulae  we  derive  the  following : 

To  object-rays,  whose  inclinations  (6)  to  the  Principal  Axis  (x)  are 
all  equal,  correspond  image-rays  which  cross  the  Focal  Plane  (ef)  of  the 
Image-Space  at  equal  distances  (k')  from  the  Focal  Point  E' ;  and, 
similarly,  to  object-rays,  which  cross  the  Focal  Plane  (<p)  of  the  Object- 
Space  at  equal  distances  (g)  from  the  Focal  Point  F,  correspond  image- 
rays  whose  inclinations  (0')  to  the  Principal  Axis  (x')  are  all  equal. 

We  had  already  perceived  (§  167)  that  to  a  bundle  of  parallel  rays 
of  one  Space-System,  say,  S,  there  corresponds  a  bundle  of  non- 
parallel  rays  of  the  other  Space-System,  the  vertex  of  which  lies  in 
the  Focal  Plane  of  2'.  We  see  now  that  this  fact  is  merely  a  particu- 
lar case  of  a  more  general  law  of  optical  imagery,  as  given  in  the  above 
statement.  The  absolute  value  of  the  focal  distance  of  the  point  R 
or  S',  where  the  object-ray  or  image-ray  crosses  the  Focal  Plane  (p 
or  e',  depends  only  on  the  magnitude  of  the  inclination  0'  or  0  of  the 
conjugate  ray  to  the  x'-  or  #-axis,  respectively. 

178.  The  Focal  Lengths/  and  e'.  Equations  (112)  obtained  in 
the  last  section,  which  may  be  written: 


tan  0'       b  '     tan  0 

afford  us  a  new  way  of  defining  the  Image-Constants  a  and  b.  Thus, 
the  constant  b  may  be  defined  as  the  ratio  of  the  Focal  distance  kf 
of  the  point  where  an  image-ray  crosses  the  Focal  Plane  of  the  Image- 
Space  to  the  tangent  of  the  angle  of  inclination  0  of  the  corresponding 
object-ray  to  the  Principal  Axis  x  of  the  Object-Space;  and,  similarly, 


§  178.]  The  Geometrical  Theory  of  Optical  Imagery.  233 

the  magnitude  a/b  may  be  defined  as  the  ratio  of  the  Focal  distance 
g  of  the  point  where  an  object-ray  crosses  the  Focal  Plane  of  the 
Object-Space  to  the  tangent  of  the  angle  of  inclination  6'  of  the  cor- 
responding image-ray  to  the  Principal  Axis  xf  of  the  Image-Space. 
From  the  equations  above,  as  well  as  from  the  Image-Equations  them- 
selves (§  174),  it  is  apparent  that  the  dimensions  of  the  Image-Constants 
a  and  b  are  different;  thus,  whereas  b  denotes  a  length,  a  denotes  an 
area.  For  this  and  other  reasons  it  is  convenient  to  introduce  at  this 
point  a  new  pair  of  symbols  /  and  e'  instead  of  a  and  &,  and  to  write  : 

/-Si     e'  =  -b.  (113) 

Thus,  the  Image-Constants  denoted  by  /  and  e'  will  be  defined  by 
the  following  formulae: 


The  constants  /  and  e'  are  called  the  Focal  Lengths  of  the  optical 
system.  According  to  ABBE,  the  proper  definitions  of  these  character- 
istic constants  of  the  optical  system  are  given  only  by  formulae  (114). 
So  soon  as  the  magnitudes  denoted  by  /  and  er  are  ascertained,  the 
optical  system  may  be  regarded  as  completely  determined. 

The  definition  of  the  Focal  Lengths  of  a  system  of  lenses,  as  given 
by  GAUSS/  is  essentially  the  same  as  ABBE'S  definition  by  means  of 
the  above  equations;  thus: 

The  Focal  Length  of  the  Object-Space  (denoted  here  by  /)  is  equal 
to  the  ratio  of  the  linear  magnitude  of  an  image  formed  in  the  Focal 
Plane  of  the  Image-Space  to  the  apparent  (or  angular)  magnitude  of  the 
corresponding  infinitely  distant  object;  and 

The  Focal  Length  of  the  Image-Space  (denoted  by  ef)  is  equal  to  the 
ratio  of  the  linear  magnitude  of  an  object  lying  in  the  Focal  Plane  of 
the  Object-  Space  to  the  apparent  magnitude  of  its  infinitely  distant  image. 

Introducing  the  Focal  Lengths  /  and  e'  ',  we  may  now  write  the 
Image-Equations  (in)  as  follows: 


. 

Provided  we  adhere  to  the  choice  of  the  positive  directions  of  the 
axes  of  co-ordinates  which  was  made  in  §  172  (where  we  had  a  <  o, 

1  See  S.  CZAPSKI:    Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  p.  40. 


234  Geometrical  Optics,  Chapter  VII.  [  §  179. 

b  >  o),  we  shall  have  always: 

/  >  o,     e'  <  o. 

179.    The  Magnification-Ratios  and  their  Relations  to  one  another. 

1 .  The  Lateral  Magnification  Y.     This,  as  has  been  already  defined 
(§  170),  is  the  ratio  of  conjugate  line-segments  lying  in  planes  at  right 
angles  to  the  Principal  Axes.     Thus, 

/  f  V1 

F  =  — =  -  =  -7;  (116) 

y       x       e' 

whence  we  see  that  the  Lateral  Magnification  F  may  have  any  value 
from  —  oo  to  +  oo ,  depending  on  the  value  of  the  abscissa  x. 

2.  The  Axial  or  Depth-Magnification    X.     By  differentiating  the 
abscissa-equation 

xx'  =  fe', 

we  obtain  for  the  ratio  of  infinitely  small  conjugate  line-segments  dx, 
dx'  of  the  Principal  Axes: 

dx  x2          fe'  * 

This  ratio,  denoted  by  X,  is  called  the  Axial  or  Depth- Magnifica- 
tion. It  is  inversely  proportional  to  the  square  of  the  abscissa  x. 
If  we  choose  the  positive  directions  of  the  axes  of  co-ordinates  so  that 
/  >  o,  e'  <  o  (see  §  178),  then  X  will  be  necessarily  positive,  and  may 
have  any  value  comprised  between  o  and  +  oo. 

Comparing  formulae  (116)  and  (117),  we  obtain  the  following  rela- 
tion between  the  Axial  Magnification  (X)  and  the  Lateral  Magnifi- 
cation (F): 

X  e'  .       . 

72  =-7;  (n8) 

and,  hence,  we  can  say:  At  each  point  the  Axial  Magnification  is  pro- 
portional to  the  square  of  the  Lateral  Magnification. 

3.  The  Angular  Magnification  Z.     Let  M,  M'  (Fig.  91)  designate 
the  positions  of  two  axial  conjugate  points,  whose  abscissae  with  respect 
to  the  Focal  Points  F,  E'  are  denoted  by  x,  x',  respectively;  so  that 

FM  =  x,     E'M'  =  x'. 

Let  the  straight  line  MR  represent  an  object-ray  crossing  the  Focal 
Plane  of  the  Object-Space  at  the  point  R  and  making  with  the  Prin- 
cipal Axis  x  of  the  Object-Space  an  angle  xMR  =  B.  Let  Sf  designate 


§  179.]  The  Geometrical  Theory  of  Optical  Imagery.  235 

the  point  where  the  conjugate  ray  S'M'  crosses  the  Focal  Plane  of 
the  Image-Space,  and  let  0'  =  LE'M'S'  denote  the  inclination  of 
this  image-ray  to  the  Principal  Axis  x'  of  the  Image-Space.  Putting 

FR  =  g,     E'S'  =  kf, 

we  have,  in  accordance  with  our  agreement  in  §  177  concerning  the 
signs  of  the  angles  0,  0' : 

g                         k' 
tan  0  =  —  - ,     tan  0  = : . 

x  x' 

The  Focal  Lengths/  and  ef,  by  definition,  are  given  by  the  formulae: 

/-  *'     -'-  g 


tan.0'  tan0'' 

And,  hence,  if  Z  denotes  the  ratio  of  the  tangents  of  the  angles  of 
inclination  to  the  Principal  Axes  of  a  pair  of  conjugate  rays  in  any  two 
conjugate  Meridian  Planes,  we  have: 

tan^          x  / 

Z  =  tanT  =  -.'=  ~?;  <"9> 

whence  it  will  be  seen  that  Z  is  independent  of  the  values  of  0,  0' 
themselves;  so  that  for  a  given  value  of  x,  the  ratio  denoted  by  Z 


FIG.  91. 
ANGULAR  MAGNIFICATION. 


,    tan0'  =  —  EfSlE'M',    Z=  tan  0'  I  tan  9, 
where  ^denotes  the  angular  magnification  for  the  conjugate  axial  points  M,  Mr  . 

has  a  constant  value.     Thus,  for  all  rays  which  pass  through  the  axial 
point  M,  the  ratio  tan  0'  :  tan  0  is  constant. 

This  ratio  denoted  by  Z  is  called  the  Angular  Magnification,  or 
the  "  Convergence-  Ratio",  and  is  an  important  magnitude  in  the  theory 
of  optical  instruments. 


236  Geometrical  Optics,  Chapter  VII.  [  §  180. 

Comparing  the  values  of  the  Magnification-Ratios  X,  Y  and  Z, 
as  given  by  formulae  (116),  (117)  and  (119),  we  have  the  following 
relations  between  them: 


=  _,    .-,    •-,         . 

180.  The  Cardinal  Points  of  an  Optical  System.  As  we  see  from 
formulae  (116)  and  (119),  the  Magnification-Ratios  Fand  Z  may  have 
any  values  comprised  between  —  oo  and  +  oo,  depending  on  the 
value  of  the  abscissa  x\  whereas  the  Depth-Magnification  X,  as  is 
shown  by  formulae  (117),  may  have  any  value  between  o  and  +  oo  ; 
since  we  assume  in  this  discussion  that  the  positive  directions  of  the 
axes  of  x,  x'  are  so  chosen  that  the  Focal  Lengths  /  and  e'  have  always 
opposite  signs.  Each  of  these  ratios  is  a  function  of  the  abscissa  x, 
so  that  by  assigning  any  particular  value  to  one  of  these  ratios,  we 
shall  thereby  determine  at  least  one  pair  of  conjugate  axial  points. 
Those  pairs  of  conjugate  axial  points  for  which  one  or  other  of  the 
magnitudes  denoted  by  X,  F,  Z  has  the  absolute  value  unity  are  all 
of  more  or  less  interest,  and  certain  of  them  are  especially  distin- 
guished in  the  theory  of  optics.  They  may  be  enumerated  in  the 
following  order: 

I.  The  two  pairs  of  conjugate  axial  points  for  which  the  Depth- 
Magnification  X  has  the  value  +  i  ;  for,  since  X  is  a  function  of  x2, 
we  shall  obtain  always  for  a  given  value  of  X  two  equal  and  opposite 
values  of  the  abscissa  x.  Thus,  putting  X  =  +  i  in  formulas  (117), 
we  find: 


so  that  there  are  two  pairs  of  conjugate  points  on  the  Principal  Axes 
of  the  optical  system  for  which  an  infinitely  small  displacement  dx 
of  the  object-point  will  correspond  to  an  equal  displacement  dx'  of 
the  image-point.  Moreover,  it  will  be  remarked  that  the  Focal  Points 
F  and  E'  are  midway  between  the  two  axial  object-points  and  the  two 
axial  image-points,  respectively.  However,  these  two  pairs  of  axial 
conjugate  points  are  of  slight  importance,  and  need  not  detain  us  any 
longer,  except  merely  to  add  that  the  Lateral  and  Angular  Magnifi- 
cations at  these  points  are  equal.  Thus,  we  have: 

F  =  Z=  ±1/7/7'. 
2.  The  most  important  and  the  most  celebrated  of  all  these  pairs 


§  180.] 


The  Geometrical  Theory  of  Optical  Imagery. 


237 


of  conjugate  axial  points  is  the  pair  named  by  GAUSS  1  the  Principal 
Points  (see  §  139)  of  the  optical  system,  which  in  our  diagrams  will 
be  designated  by  the  letters  A  and  A'  (Fig.  92).  The  Principal 


N 


FIG.  92. 


CARDINAL  POINTS  OF  OPTICAL  SYSTEM.    Focal  Points  F,  E1 ';  Principal  Points  A,  A';  Nodal 
Points  N,  N'. 

FA=N'E'=f;    EfA'=NF=ef;    NA—'N' A' ;    AV=A'V;     L  ANV=9  =  L  A' N' V  =  6'. 

Points  are  denned  by  the  value  Y  =  +  I.  Putting  Y  =  y'/y  =  +  i 
in  the  equations  (116),  we  obtain  for  the  abscissae  of  the  Principal 
Points: 


FA  =  /, 


=  E'A'  =  e' 


(131) 


and,  hence,  The  Focal  Lengths  /,  ef  of  an  optical  system  may  also  be 
defined  as  the  abscissa,  with  respect  to  the  Focal  Points  F,  Er,  of  the 
Principal  Points  A,  A',  respectively. 

If  the  positive  directions  of  the  Principal  Axes  are  determined  by 
the  directions  pursued  by  the  light  in  traversing  these  lines,  then,  as 
has  been  repeatedly  stated,  /  will  be  positive  and  e'  negative  (see 
§  178);  hence,  the  Primary  Principal  Point  A  will  lie  always  on  the 
positive  half  of  the  #-axis,  and  the  Secondary  Principal  Point  A'  will 
lie  on  the  negative  half  of  the  #'-axis. 

The  pair  of  conjugate  planes  at  right  angles  to  the  Principal  Axes 
at  the  Principal  Points  A,  A'  were  likewise  named  by  GAUSS  the 
Principal  Planes  of  the  system.  These  planes  are  characterized  by 
the  fact  that  to  any  point  V  in  the  Principal  Plane  of  the  Object- 
Space  there  corresponds  a  point  V  in  the  Principal  Plane  of  the  Image- 
Space,  such  that  AV  =  A'V  ',  so  that  an  object  lying  in  the  Primary 
Principal  Plane  will  be  portrayed  by  an  image  lying  in  the  Second- 
ary Principal  Plane,  which  is  equal  to  the  object  in  every  particular. 

We  may  remark  also  that  at  the  Principal  Points  x  =  /,  x'  =  e' 
we  have  also: 


1  C.  F.  GAUSS:  Dioptrische  Untersuchungen  (Goettingen,  1841),  \  7. 


238  Geometrical  Optics,  Chapter  VII.  [  §  180. 

The  points  A,  A'  are  sometimes  called  also  the  Positive  Principal 
Points  in  order  to  distinguish  them  from  another  pair  of  axial  conju- 
gate points  called  by  TOEPLER  1  the  Negative  Principal  Points,  which 
are  defined  by  the  value  Y  =  —  i.  These  points  are,  however,  of 
no  particular  importance. 

3.  The  conjugate  axial  points  TV,  TV',  for  which  the  Angular  Magni- 
fication has  the  value  Z  =  tan  6'  :  tan  6  =  +  1  ,  were  named  by  LISTING  2 
the  Nodal  Points  of  the  system.  These  points,  which  are  next  in 
importance  to  the  Principal  Points,  are  characterized  by  the  following 
property  : 

To  an  object-ray  crossing  the  x-axis  at  the  Primary  Nodal  Point  N 
at  an  inclination  8  there  corresponds  an  image-ray  crossing  the  x'-axis 
at  the  Secondary  Nodal  Point  N'  at  an  inclination  0'  =  6. 

Putting  Z  =  +  i  in  formulae  (119),  we  find  for  the  abscissae,  with 
respect  to  the  Focal  Points  F,  E'  ,  of  the  Nodal  Points  N,  N': 

x  =  FN  =  -  e',     x'  =  E'N'  =  -  /; 
or  (Fig.  92)  : 

FA  =  N'E'  =/;     E'A'  =  NF  =  e'.  (122) 

Moreover,  since 


=  AF+FN=  -(/  +  O,     A'N'  =  A'E'_+  E'N'  =  •- 

we  have:( 

AN  =  A'N'.  (123) 

Hence,  the  two  Nodal  Points  are  equidistant  from  the  Principal  Points; 
and,  since  the  abscissae  of  TV,  N',  with  respect  to  A,  A',  respectively, 
have  the  same  sign,  the  Nodal  Points  lie  always  either  both  to  the 
right  or  both  to  the  left  of  the  corresponding  Principal  Points.  And 
if  A  N  =  o,  then  Af  N'  =  o  also. 
For  Z  =  +  i,  we  have: 

' 


The  planes  perpendicular  to  the  Principal  Axes  at  the  points  TV,  TV' 
are  called  the  Nodal  Planes  of  the  system.  TOEPLER  likewise  dis- 
tinguished a  pair  of  Negative  Nodal  Points  defined  by  Z  =  —  i. 

These  distinguished  pairs  of  conjugate  axial  points  are  called  the 

1  A.  TOEPLER:  Bemerkungen  ueber  die  Anzahl  der  Fundamentalpuncte  eines  beliebigen 
Systems  von  centrirten  brechenden  Kugelflaechen  :  POGG.  Ann.,  cxlii.   (1871),  232-251. 

2  J.   B.  LISTING:  Beitrag  zur  physiologischen  Optik:   Goettinger  Studien,   1845.     See 
also  article  by  LISTING  on  the  Dioptrics  of  the  Eye,  published  in  R.  WAGNER'S  Handwoer- 
terbuch  d.  Physiologic  (Braunschweig,  1853),  Bd.  iv.,  p.  451' 


§  181.]  The  Geometrical  Theory  of  Optical  Imagery.  239 

Cardinal  Points  of  the  optical  system  ;  and  some  writers  include  also 
under  this  designation  the  Focal  Points  F,  E'.  Knowing  the  positions 
of  one  pair  of  the  Cardinal  Points,  and  knowing  also  the  Focal  Lengths 
of  the  optical  system,  we  can  determine  completely  the  character  of 
the  imagery.1 

181.  The  Image-Equations  referred  to  a  Pair  of  Conjugate  Axial 
Points.  It  will  be  convenient  sometimes,  and  always  in  the  case  of 
Telescopic  Imagery  (Art.  50),  to  select  as  origins  of  the  two  systems 
of  co-ordinates  some  other  pair  of  axial  points  besides  the  Focal  Points 
which  have  been  used  hitherto  for  this  purpose.  Thus,  suppose  we 
take  two  conjugate  axial  points  0,  Or  as  origins,  and  let  the  co-ordi- 
nates of  the  conjugate  points  Q,  Q'  with  respect  to  0,  0'  be  denoted 
as  follows: 

OM  =  £,     MQ  =  y,     O'M'  =  £',  M'Q'  =  y'; 

where  M,  M'  are  the  feet  of  the  perpendiculars  let  fall  from  Q,  Q'  on 
the  axes  of  x,  xf,  respectively.  Moreover,  let 


x,     E'M'  =  x',     FO  =  xQ,     E'O'  =  x'Q  -, 
so  that 

xx  =  x^  =  fe. 
Now 

x  =  x0  +  £,     x  =  x'0  +  £', 
and,  therefore: 

(*o  +  £)0o  +  £')  =  xjc'o  ; 

which  may  be  written  : 

*^0     i     ^0     i  /  \ 

y  +  ^7  +  i=o;  (124) 

which  is  the  required  relation  between  the  abscissae  £  and  £'.  The 
constants  x0,  x'Q  which  occur  in  this  formula  are  the  distances  of  the 
origins  0,  0'  from  the  Focal  Points  F,  E',  respectively. 

1  In  connection  with  this  subject,  the  following  writers  may  be  consulted  (in  addition 
to  those  already  named): 

C.  G.  NEUMANN:  Die  Haupl-  und  Brenn-Punkte  eines  Linsen-Sy  stems.  Elementare 
Darstellung  der  durch  GAUSS  begruendeten  Theorie  (Leipzig,  1866). 

J.  A.  GRUNERT:  Ueber  merkwuefdige  Puncte  der  Spiegel-  und  Linsen-Systeme:  Grun. 
Arch.  f.  Math.  Phys.,  xlvii.  (1867),  84-105. 

F.  LIPPICH:  Fundamentalpunkte  eines  Systemes  centrirter  brechender  Kugelflaechen: 
Mitt,  des  naturw.  Ver.f.  Steiermark,  ii.  (1871),  429-459. 

L.  MATTHIESSEN  :  Grundriss  der  Dioptrik  geschichteter  Linsensysteme.  Mathematische 
Einleftung  in  die  Dioptrik  des  menschlichen  Auges  (Leipzig,  1877).  Also,  Ueber  eine 
Methode  zur  Berechnung  der  sechs  Cardinalpuncte  eines  centrierten  Systems  sphaerischer 
Linsen:  Zft.  f.  Math.  u.  Phys.,  xxiii.  (1878),  187-191.  Also,  Bestimmung  der  Cardinal- 
puncte eines  dioptrisch-katoptrischen  Systems  centrirter  sphaerischer  Flaechen,  mittels 
Kettenbruchdeterminanten  dargestellt:  Zft.  f.  Math.  u.  Phys.,  xxxii.  (1887),  170-175. 

The  above  is  only  a  partial  list  of  the  writers  on  this  subject. 


240  Geometrical  Optics,  Chapter  VII. 

For  the  Lateral  Magnification  at  Af,  M'  we  obtain: 

Y  =  y-  =  —f—  =  ?M-^ 
~  ' 


§181. 


(125) 


The  Angular  Magnification,  in  terms  of  the  abscissae  £,  £',  is  given 
as  follows: 

-tan*'  *0  +  £  / 

z  =  ^nT=    -T-   ~*T+T-         (I26) 

If  F0  denotes  the  Lateral  Magnification  at  the  points  0,  0',  then 
F0  =f/xQ  =  x'Q/e.  Hence,  if  we  choose,  we  may  eliminate  the  con- 
stants XQ,  x'0  in  the  above  equations  by  putting : 


—  —       Y'  —  p'V  • 

T7"     '  0  0    » 

*« 


thus, 


e'Yn 


p'Y  -J-  £' 

K   *  ft     \      S 


(127) 


In  particular,  if  the  origins  of  the  two  systems  of  co-ordinates  are 
the  pair  of  conjugate  axial  points  A,  A'  called  the  Principal  Points 
(§  1 80),  and  if  for  this  special  case  we  denote  the  abscissae  of  the  points 
Q,  Q'  by  u,  u' ,  so  that 

AM  =  u,     A'M'  =  u', 

then,  writing  u,  u'  in  place  of  £,  £',  respectively,  in  formulae  (127),  and 
putting  F0  =  i,  we  obtain  the  Image- Equations  referred  to  the  Prin- 
cipal Points  as  origins,  as  follows: 


(128) 


F  = 
Z  = 

w 

y'  / 

+  J+i=o, 
e'  +  u' 

/«' 

y  /  + 

tanfl' 

/                          ^^ 

w           e 

/+« 

e'M' 

/» 

tan5 

e'               « 

?'  +  «'  ~  u'' 

182.] 


The  Geometrical  Theory  of  Optical  Imagery. 


241 


182.     Geometrical  Constructions  of  Conjugate  Points  of  an  Optical 
System. 

i.    Construction  of  Conjugate  Axial  Points.     The  equation 


suggests  a  simple  method  of  construction  of  the  pair  of  conjugate  axial 
points  M,  M'  ,  provided  we  know  the  positions  and  directions  of  the 
Principal  Axes  x,  x',  the  positions  of  the  two  Principal  Points  A,  A', 
and  the  magnitudes  of  the  Focal  Lengths  /,  e'  of  the  Optical  System. 
For  since/  =  FA,  e'  =  E'A',  the  equation  above  may  be  written: 


AF      A'E' 
u    +    u' 


i; 


and,  hence,  if  we  suppose  the  two  Principal  Points  A,  A'  (Fig.  93)  are 
placed  in  coincidence  with  each  other  so  that  the  positive  directions 
of  the  Principal  Axes  x, 
xf  make  with  each  other 
at  A  (or  A')  any  angle 
xAx'  different  from  zero, 
and  if  through  the  Focal 
Points  F  and  Er  we  draw 


F' 


FIG.  93. 


CONSTRUCTION  OF  CONJUGATE  AXIAL  POINTS  M,  Mf 
(OR  L,  U)  OF  AN  OPTICAL  SYSTEM. 


FA 


•  u'. 


straight  lines  parallel  to 
x'  and  x,  respectively,  in- 
tersecting each  other  at  a 
point  0,  then  any  straight 
line  drawn  through  0  will 
make  on  the  axes  x, 
x'  intercepts  AM  =  u, 
A'M'  —  u'j  respectively, 
which  will  satisfy  the 

above  equation.     In  fact  the  point  0  is  the  centre  of  perspective  of 
the  two  point-ranges  #,  x' . 

2.  Construction  of  Conjugate  Points  Q,  Qf  not  on  the  Principal 
Axes.  Suppose  that  the  optical  system  is  given  by  assigning  the 
positions  and  directions  of  the  Principal  Axes  x,  xf  (Fig.  94),  the  posi- 
tions of  the  two  Focal  Points  F,  E'  and  the  Focal  Lengths/,  e1 '.  The 
Principal  Points  A,  A'  may  be  located  at  once,  since  FA  =/,  E'A '  =  e' 
(§180);  and  the  planes  through  these  points  perpendicular  to  the 
Principal  Axes  are  the  Principal  Planes.  The  point  Q'  is  the  vertex 

17 


242 


Geometrical  Optics,  Chapter  VII. 


§182. 


of  the  bundle  of  image-rays  corresponding  to  the  bundle  of  object- 
rays  whose  vertex  is  the  Object-Point  Q.  If,  therefore,  we  can  deter- 
mine two  of  the  rays  of  the  bundle  of  image-rays,  they  will  suffice  to 
determine  by  their  point  of  intersection  the  Image-Point  Qf.  Thus, 
for  example,  to  the  object-ray  Q  V  which  is  parallel  to  the  x-axis 
there  corresponds  an  image-ray  which  goes  through  the  Focal  Point 
£';  and  moreover,  this  image-ray  will  cross  the  Principal  Plane  of 
the  Image-Space  at  a  point  V  conjugate  to  the  point  V  where  the 


FIG.  94. 

CONSTRUCTION  OF  CONJUGATE  POINTS  Q,  Q'  OF  AN  OPTICAL  SYSTEM.    F  and  E'  are  the  Focal 
Points  ;  A  and  A'  are  the  Principal  Points  ;  and  TV  and  N'  are  the  Nodal  Points. 


;    EfA'  =  NF=ef;    AN=A'N'; 
/  ANU=  t-A'N'tf  ',    AV=A'V;    AU=A'U'\    AW=A'W. 

corresponding  object-ray  crosses  the  Principal  Plane  of  the  Object- 
Space,  determined,  according  to  the  property  of  the  Principal  Planes 
(§  1  80),  by  the  fact  that  AV  =  A'  V  .  Again,  the  object-ray  QW, 
which  goes  through  the  Focal  Point  F,  and  which  meets  the  Principal 
Plane  of  the  Object-Space  at  the  point  W,  must  correspond  to  an 
image-ray,  which,  crossing  the  Principal  Plane  of  the  Image-Space  at 
a  point  W  such  that  AW  =  A'W,  proceeds  parallel  to  the  Principal 
Axis  x'\  and  the  intersection  of  this  ray  with  the  other  image-ray 
V'E'  will  determine  the  Image-Point  Q'  conjugate  to  the  Object- 
Point  Q. 

If  we  know  the  positions  of  the  two  Focal  Points  F,  E',  and  if  we 
know  also  the  Focal  Lengths  /,  e',  we  may  locate  the  Nodal  Points 
N,  N'  (§  1  80).  To  an  object-ray  QN  meeting  the  Principal  Plane 
of  the  Object-Space  at  a  point  U  there  corresponds  an  image-ray, 
which,  crossing  the  Principal  Plane  of  the  Image-Space  at  a  point  Uf 
such  that  A  U  =  A1  U',  has  the  same  inclination  to  the  #'-axis  as  the 
object-ray  QN  has  to  the  #-axis;  that  is,  Z.ANU=£A'N'Uf. 
Thus,  the  point  Q'  may  be  determined  as  the  point  of  intersection  of 
any  pair  of  the  three  image-rays  V'Q',  U'Q'  and  W'Q*. 


§  183.]  The  Geometrical  Theory  of  Optical  Imagery.  243 

ART.  50.     TELESCOPIC  IMAGERY. 

183.  The  Image-Equations  in  the  Case  of  Telescopic  Imagery. 
In  the  special  and  singular  case  of  Telescopic  Imagery  (§  165),  the 
Image-Equations  (no)  referred  to  the  Focal  Points  F,  E'  are  not 
applicable,  because  in  this  case  the  Focal  Planes  (p  and  e'  are  no  longer 
actual,  or  finite,  planes,  but  they  are  the  infinitely  distant  planes  e 
and  (f>  of  the  Object-Space  S  and  the  Image-Space  2',  respectively. 
The  infinitely  distant  planes  are  not  only  the  Focal  Planes,  but  they 
are  also  a  pair  of  conjugate  planes;  so  that,  if  we  were  consistent  in 
our  notation,  and  if  we  designated  the  Focal  Plane  of  the  Object- 
Space  by  (p,  in  the  case  of  Telescopic  Imagery  we  should  designate 
the  Focal  Plane  of  the  Image-Space  by  <p'.  In  the  language  of  geom- 
etry the  two  Space-Systems  S,  S'  are  said  to  be  in  "affinity"  with 
each  other.  Each  pair  of  conjugate  plane-fields  ir,  TT'  of  S,  S'  are 
also  in  affinity  with  each  other,  because  to  every  infinitely  distant 
straight  line  of  S  there  corresponds  an  infinitely  distant  straight  line 
of  2'.  Thus,  also,  each  pair  of  conjugate  point-ranges  of  2  arid  2' 
are  "protectively  similar",  so  that  corresponding  segments  of  them 
are  in  a  constant  ratio  to  each  other  (§  166).  Hence,  the  image  in 
2'  of  a  parallelogram  of  2  will  likewise  be  a  parallelogram;  and  so, 
also,  the  image  of  a  parallelepiped  will  be  a  parallelepiped.  To  a 
bundle  of  parallel  object-rays  will  correspond  a  bundle  of  parallel 
image-rays. 

Since  corresponding  point-ranges  are  "protectively  similar",  we 
can  say: 

In  the  case  of  Telescopic  Imagery,  the  Magnification- Ratio  has  the 
same  value  for  all  parallel  rays. 

This  fundamental  characteristic  of  Telescopic  Imagery  will  enable 
us  to  deduce  the  Image-Equations  immediately.  Thus,  selecting  as 
origins  of  the  two  systems  of  co-ordinates  any  pair  of  conjugate  points 
0,  0',  let  us  take  as  axes  of  x,  y,  z  any  three  straight  lines  meeting  in 
0,  and  let  the  three  straight  lines  conjugate  to  x,  y,  z  which  meet  in 
0'  be  selected  as  axes  of  x',  y',  z'.  If  the  magnification-ratios  for  the 
three  bundles  of  parallel  object-rays  to  which  the  axes  of  x,  y,  z  belong 
are  denoted  by  the  constants  p,  q,  r,  we  may  write  the  I  mage- Equations 
for  the  case  of  Telescopic  Imagery,  as  follows : 

x'  =  px,     y'  =  qy,     z'  =  rz.  (129) 

If  we  select  a  set  of  rectangular  axes  in  the  Object-Space,  the  axes  of 
x',  y',  z'  in  the  Image-Space  will  in  general  be  oblique.     But  in  the 


244  Geometrical  Optics,  Chapter  VII.  [  §  184. 

two  projective  bundles  of  rays  O,  0',  there  is  always  one  set  of  mutually 
perpendicular  rays  of  0  to  which  corresponds  also  a  system  of  three 
mutually  perpendicular  rays  of  0' ';  so  that  if  we  choose  these  two 
particular  sets  of  corresponding  rays  as  axes  of  co-ordinates  of  the 
two  Space-Systems,  the  equations  above  will  be  the  general  Image- 
Equations,  referred  to  rectangular  axes,  for  the  case  of  Telescopic 
Imagery. 

It  will  be  remarked  that  in  the  general  case  of  Telescopic  Imagery 
the  Image-Equations  involve  at  least  three  independent  constants. 
A  striking  difference  between  Telescopic  Imagery  and  Optical  Imagery 
in  general  is  to  be  noted  in  the  fact  that  in  the  former  there  are  no 
Principal  Axes;  so  that  it  is  merely  a  matter  of  preference  which  of 
the  axes  of  co-ordinates  is  selected  as  the  axis  of  x. 

However,  if  (as  is  practically  nearly  always  the  case)  the  Imagery 
is  symmetrical  with  respect  to  one  pair  of  the  conjugate  axes  of  the 
two  systems  of  rectangular  co-ordinates,  it  is  usual  to  select  these 
as  the  axes  of  x  and  x' .  In  this  case  putting  r  =  q,  we  may  write 
the  Image-Equations  as  follows: 

x'  =  px,     y'  =  qy,     z'  =  gz.  (130) 

184.  Characteristics  of  Telescopic  Imagery.  In  the  case  of  Tele- 
scopic Imagery,  both  the  Lateral  Magnification  Y  and  the  Depth- 
Magnification  X  are  constant.  Thus: 

V  dxf      y.' 

' 


In  regard  to  the  Focal  Lengths  /  and  e',  defined  as  in  §  178,  it  is 
obvious  that  we  have  here : 

/  =  e'  =  oo. 

But  by  formula  (118)  we  have  X/Y2  =  —  e'/f;  hence,  here: 

'•-"      .       J=~?~  (I32) 

Accordingly,  we  may  say  that  the  characteristic  of  Telescopic  Imagery 
is  that,  whereas  the  Focal  Lengths  f  and  e'  are  infinite,  the  ratio  of  the 
Focal  Lengths  is  finite. 

Introducing  this  finite  ratio  as  one  of  the  image-constants  and  the 
Lateral  Magnification  Y  =  q  =  F0  as  the  other  constant,  we  may 
write  the  Image-Equations  for  the  case  of  Telescopic  Imagery  as 


§  185.]  The  Geometrical  Theory  of  Optical  Imagery.  245 

follows  : 

x'  e'       yf 

-=-^07,    ~  =  Y,  (133) 

The  Angular  Magnification  Z  is  given  by  the  formula  : 

tan  8'       Y 
tan  (9   ~  X  ' 

Hence,  for  the  case  of  Telescopic  Imagery: 


p  e'Y0' 

Thus,  the  Angular  Magnification  in  Telescopic  Imagery  is  constant 
also.  It  may  be  remarked  that  the  Angular  Magnification  is  an 
especially  important  magnitude  in  this  kind  of  imagery;  for  when  we 
are  considering  the  infinitely  distant  image  of  an  infinitely  distant 
object,  the  Angular  Magnification  is  the  only  kind  of  magnification 
that  conveys  any  meaning.  If  in  the  Image-Equations  we  introduce 
Z  =  9.1  P  =  ZQ  and  the  ratio  er  If  =  —  p/q2  as  the  two  image-con- 
stants, these  equations  may  also  be  expressed  as  follows: 

f  l    y'      f  i 


ART.  51.     COMBINATION  OF  TWO  OPTICAL  SYSTEMS. 

185.  The  Problem  in  General.  A  series  of  Optical  Systems  may 
be  so  arranged  one  after  the  other  that  the  Image-Space  of  one  system 
is  at  the  same  time  the  Object-Space  of  the  following  system,  and  so  on. 
The  resultant  effect  of  all  these  successive  imageries  will  be  an  imagery 
which  may  be  regarded  as  due  to  a  single  optical  system  which  by 
itself  would  produce  the  same  effect.  An  optical  instrument,  whose 
function  is  to  produce  an  image  of  an  external  object,  is,  in  fact,  nearly 
always  a  compound  system  or  combination  of  simpler  systems.  Pro- 
vided we  know  the  Focal  Lengths  and  the  positions  of  the  Focal 
Points  and  of  the  Principal  Axes  of  each  of  the  component  systems, 
it,  is  always  possible  to  ascertain  the  Focal  Lengths  and  the  positions 
of  the  Focal  Points  of  the  compound  system.  In  case  the  system  is 
composed  of  spherical  refracting  (or  reflecting)  surfaces  with  their 
centres  ranged  along  a  straight  line,  the  Principal  Axes  of  each  of 
the  elements  of  the  system  will  coincide  with  the  "optical  axis"  (§  135). 
This  is  usually  the  case  in  an  actual  optical  instrument,  and  the 


246 


Geometrical  Optics,  Chapter  VII. 


[  §  185. 


problem  is  greatly  simplified  by  this  condition.  However,  following 
CzAPSKi,1  and  supposing  at  first  that  we  have  only  two  component 
systems,  we  shall  consider  here  a  rather  more  general  case  than  the 
one  above-mentioned.  Thus,  the  only  restriction  which  we  shall 
make  is  the  following: 

The  Principal  Axis  (x{)  of  the  Image-Space  of  System  (I)  shall  be  also 
the  Principal  Axis  (x2)  of  the  Object- Space  of  System  (II}. 

(Since  this  condition  is  usually  satisfied  in  the  case  of  imagery  by 
means  of  narrow  bundles  of  rays  inclined  to  the  Principal  Axes  at 
finite  angles,  the  results  which  we  shall  obtain  here  will  be  directly 
applicable  also  to  this  case,  as  we  shall  have  occasion  of  seeing  in 
a  subsequent  chapter.  See  §  248.) 

Let  Flf  E(  and  F2,  E2  (Fig.  95)  designate  the  positions  of  the  Focal 
Points  of  the  systems  (I)  and  (II),  respectively;  and  let/!,  e{  and/2,  e'2 


FIG.  95. 

COMBINATION  OF  Two  OPTICAL  SYSTEMS,  fi  and  E{  and  Fz,  Ezf  mark  the  positions  of  the  Focal 
Points,  and  Ai,  Ai'  and  A%,  Ad  the  positions  of  the  Principal  Points  of  systems  (I)  and  (II),  re- 
spectively; and  F,  Er  mark  the  positions  of  the  Focal  Points  of  the  compound  system  (I  +  11). 
/i  =»  FiAi,  ei'  =  Ei'Ai',  and/a  =  F*Az,  <?a'  =  E<£  '  Az'  denote  the  Focal  lengths  of  systems  (I)  and  (ll). 
respectively;  whereas/,  e'  denote  the  Focal  lengths  of  the  compound  system.  The  "interval" 
between  the  two  systems  (l)  and  (II)  is  Ei'Fs  =  A. 


denote  the  Focal  Lengths  of  the  two  systems.  Since  we  have  assumed 
that  the  Principal  Axis  (x()  of  the  Image-Space  of  (I)  is  coincident 
with  the  Principal  Axis  (x2)  of  the  Object-Space  of  (II),  the  Focal 
Planes  at  E[  and  F2  will  be  parallel.  The  relative  position  of  the 
two  component  systems  may  be  assigned  by  giving  the  distance  of 
the  point  F2  from  the  point  E'lt  that  is,  the  abscissa  of  F2  with  respect 
to  E{.  This  magnitude,  usually  denoted  by  writers  on  Optics  by  the 
symbol  A,  that  is, 

A  =£;/?„ 

and  reckoned  positive  or  negative  according  as  the  direction  from  E( 

1  S.  CZAPSKI:  Theorie  der  optischen  Instrumente,  nach  ABBE  (Breslau,  1893),  pages  46, 
foil. 


§  185.]  The  Geometrical  Theory  of  Optical  Imagery.  247 

to  F2  is  the  same  as  or  opposite  to  that  along  which  the  light  travels, 
is  called  the  interval  between  the  two  systems. 

The  Focal  Points  of  the  compound  system  will  be  denoted  by  the 
letters  F  and  E'  (without  any  subscripts) ;  and,  similarly,  the  Focal 
Lengths  of  the  compound  system  will  be  denoted  by  the  symbols  / 
and  e' . 

The  problem  is,  therefore,  with  the  data  above-mentioned  denning 
the  component  systems,  to  determine  the  Focal  Points  and  the  Focal 
Lengths  of  the  compound  system. 

In  the  first  place,  it  is  obvious  that  the  Focal  Planes  at  the  points 
F  and  Fl  are  parallel,  as  is  also  the  case  with  the  Focal  Planes  at  the 
points  E'  and  E'2.  For  in  the  system  (I),  to  the  sheaf  of  planes  which 
are  parallel  to  the  Focal  Plane  at  ^  there  corresponds  a  sheaf  of 
planes  which  are  parallel  to  the  Focal  Plane  at  E(  (§  166),  and  which 
are  therefore,  by  hypothesis,  parallel  likewise  to  the  Focal  Plane  at 
F2;  and  to  this  sheaf  of  planes  in  the  Object-Space  of  system  (II) 
there  corresponds  a  sheaf  of  planes  which  are  parallel  to  the  Focal 
Plane  at  E'2.  Hence,  to  planes  which  are  parallel  to  the  Focal  Plane 
at  Fl  there  correspond  planes  which  are  parallel  to  the  Focal  Plane 
at  E2.  But  we  have  seen  that  in  an  optical  system,  in  general,  the 
only  two  sheaves  of  parallel  planes  which  are  conjugate  sheaves  of 
planes  are  the  sheaves  to  which  the  Focal  Planes  belong.  Hence, 
the  Focal  Planes  at  F  and  E'  are  parallel  to  the  Focal  Planes  at  Fl 
and  E2,  respectively. 

Moreover,  the  Principal  Axis  (jq)  of  the  Object-Space  of  system  (I) 
is  also  the  Principal  Axis  (x)  of  the  Object-Space  of  the  compound 
system;  and,  similarly,  the  Principal  Axis  (x'2)  of  the  Image-Space  of 
system  (II)  is  also  the  Principal  Axis  (x')  of  the  Image-Space  of  the 
compound  system.  For  since  x±  and  x'  are  obviously  a  pair  of  conju- 
gate rays  with  respect  to  the  compound  system,  and  since  the  Principal 
Axes  of  an  optical  system  have  been  defined  (§  167)  as  that  pair  of 
conjugate  rays  which  meet  at  right  angles  the  Focal  Planes  of  the  sys- 
tem, it  follows  that  x±  and  x2  must  coincide  with  x  and  x',  respectively. 

We  proceed,  in  the  next  place,  to  ascertain  the  positions  of  the  Focal 
Points  F,  E'  of  the  compound  system.  Consider,  for  example,  an  object- 
ray  proceeding  in  the  direction  QVl  parallel  to  the  Principal  Axis 
(x)  of  the  Object-Space.  Emerging  from  the  first  system,  this  ray 
will  go  through  the  Focal  Point  E{  of  the  Image-Space  of  this  system, 
and,  traversing  the  second  system,  will  finally  cross  the  Principal  Axis 
(x')  of  the  Image-Space  of  the  compound  system  at  the  point  E' \ 
so  that,  with  respect  to  system  (II),  the  points  E{  and  E'  are  a  pair  of 


248  Geometrical  Optics,  Chapter  VII.  [  §  185. 

conjugate  points,  and  therefore,  according  to  the  first  of  formulae  (115): 

that  is, 

A 

Since  we  know  the  position  of  the  point  E2,  this  equation  enables  us 
to  determine  that  of  the  Focal  Point  Er  of  the  Image-Space  of  the 
compound  system. 

Again,  consider  an  object-ray  QF  going  through  the  Focal  Point  F 
of  the  Object-Space  of  the  compound  system.  This  ray,  after  travers- 
ing the  entire  system,  must  emerge  parallel  to  the  Principal  Axis  (#') 
of  the  Image-Space  of  the  compound  system.  And,  therefore,  it  must 
have  passed  through  the  Focal  Point  F2  of  the  Object-Space  of  system 
(II),  and,  hence,  the  points  F  and  F2  must  be  a  pair  of  conjugate  axial 
points  with  respect  to  system  (I).  Accordingly,  in  the  same  way 
as  above : 

77   77        77'  77  f  * 

*\*  '  -^1^2  s*/i*n 
that  is, 


whereby  the  position  of  the  Focal  Point  F  of  the  Object-Space  of  the 
compound  system  can  be  located  with  respect  to  the  position  of  the 
given  point  F±. 

Thus,  having  located  the  positions  of  the  Focal  Points  F,  Ef  of  the 
compound  system,  we  have  next  to  determine  the  Focal  Lengths  /,  ef. 
Recalling  the  definitions  of  the  Focal  Lengths  as  given  by  formulae 
(114),  viz.: 


tan0' 

let  us  consider  again  the  two  rays  which  we  have  already  employed. 
The  ray  QV19  which  proceeds  parallel  to  the  Principal  Axis  (x)  of 
the  Object-Space  of  the  compound  system,  crosses  the  Focal  Plane  v 
at  the  height  g  =  AjV^  and  after  traversing  the  entire  system, 
emerges  so  as  to  cross  the  Principal  Axis  (*')  of  the  Image-Space  at 
the  Focal  Point  £'  at  an  angle  0'  =  ^A2E'V2,  so  that  e'  =  g/tan  6'. 
This  ray  crosses  the  Principal  Axis  (x{)  of  the  Image-Space  of  sys- 
tem (I)  at  the  point  E{  (Fig.  95),  so  that 

•=«•  tan 


§  185.J  The  Geometrical  Theory  of  Optical  Imagery.  249 

And  since  E[  and  Ef  are  a  pair  of  conjugate  axial  points  with  respect 
to  system  (II),  the  ratio  tan  0'  :  tan  /.A(E(V(  is  the  value  of  the 
Angular  Magnification  (Z2)  of  system  (II)  for  this  pair  of  conjugate 
points.  Applying,  therefore,  formula  (119),  we  obtain: 

tan0'         _       F2E{  _  A 
tan  Z  A  (E't  V(  e'2     ~  e'2 ' 

Thus,  the  formula  e'  =  g/tan  0'  becomes: 

,_e(e, 
'    A    ' 

whereby  the  magnitude  of  the  Focal  Length  er  of  the  Image-Space  of 
the  compound  system  is  determined  in  terms  of  the  known  magnitudes 
elt  e'2  and  A. 

Similarly,  if  /.A^FX^  =  0  is  the  inclination  to  the  #-axis  of  the 
object-ray  QXlf  which  goes  through  the  Focal  Point  F  of  the  Object- 
Space,  and  which,  after  traversing  the  entire  system,  emerges  in  the 
direction  X'2Q'  parallel  to  the  #'-axis,  and  if  kf  denotes  the  height  at 
which  the  emergent  ray  crosses  the  Focal  Plane  e'  of  the  Image-Space, 
then  /  =  k' /tan  0.  This  ray  crossed  the  Principal  Axis  (x2)  of  the 
Object-Space  of  system  (II)  at  the  point  F2,  so  that 

£'  =/2-  tan^A2F2X2. 

The  points  F  and  F2  are  a  pair  of  conjugate  axial  points  with  respect 
to  system  (I),  and  the  value  of  the  Angular  Magnification  (Zx)  of  this 
system  for  this  pair  of  conjugate  points  is  given  by  the  ratio 

tan  £A2F2X2  :  tan  0. 
Thus,  by  formula  (119),  we  obtain: 

tan  Z.A2F2X2  _          f,        _£ 
tan0  E(F2~      A* 

Accordingly,  we  find: 

f  4/2 

/=     "   A    ; 

whereby  the  magnitude  of  the  Focal  Length  /  of  the  Object-Space  of 
the  compound  system  can  be  determined  in  terms  of  the  known 
constants /!,/2  and  A. 

The  formulae  for  the  Focal  Lengths  /  and  ef  may  be  obtained  also 
by  considering  the  ray,  which,  proceeding  from  the  Object-Point  (), 


250 


Geometrical  Optics,  Chapter  VII. 


[  §  185. 


goes  through  the  Focal  Point  Ft  of  the  Object-Space  of  system  (I), 
and  which,  therefore,  emerges  from  system  (II)  so  as  to  cross  the  x'-axis 
at  the  Focal  Point  Ef  of  the  Image-Space  of  system  (II).  Thus,  from 
Fig.  95,  we  have: 


/i 


A\W\ 


A2W2 


tan 


tan  Z  A'2E'2W2  * 


Now  A{W/l  =  A2W2,  and  therefore: 


tan 


tan 


But  since  Fl  and  E2  are  conjugate  axial  points  with  respect  to  the 
compound  system,  we  have  by  formula  (119): 


tan  ^A2E2W2 
tan 


} 


FF\ 

e'   • 


Hence : 


and,  therefore,  as  before 


FF, 

e'   ' 


/1/2  f  _   ei€2 

*        '  ' 


A  '  A 

The  formulae  derived  above  may  be  collected  as  follows  : 
Positions  of  the  Focal  Points  F,  E': 

77  77          fiei  77'  77'          J2C2 

FJ:-—,     E2E  -r—  ; 
Magnitudes  of  the  Focal  Lengths  f,  e': 

f  _         JIJ2  ,          eie2 

}~~    A'     e        A  > 


(136) 


The  influence  of  the  interval  A  between  the  two  systems,  which 
forms  the  denominator  of  the  right-hand  member  of  each  of  these  for- 
mulae, is  at  once  apparent.  Two  given  systems  (I)  and  (II)  may  be 
combined  in  an  infinite  number  of  ways  by  merely  altering  the  inter- 
val A  either  as  to  its  magnitude  or  as  to  its  sign  or  as  to  both.  So 
long  as  this  magnitude  A  is  different  from  zero,  and  none  of  the  Focal 
Points  of  the  component  systems  are  situated  at  infinity,  we  shall  have 
a  compound  system  with  finite  Focal  Lengths  /,  e'. 


§  186.] 


The  Geometrical  Theory  of  Optical  Imagery. 


251 


186.  Special  Cases  of  the  Combination  of  Two  Optical  Systems. 
We  may  consider  several  special  cases  of  the  combination  of  two  optical 
systems  as  follows: 

i.  The  Case  when  the  "Interval"  is  zero  (A  =  o),  the  Focal  Lengths 
/u  e(  and  /2,  e'2  of  the  component  systems  (I)  and  (II)  being  all  finite. 

In  this  case  the  compound  system  will  be  telescopic,  since  according 
to  equations  (136),  we  have  here  /  =  e'  =  oo,  whereas  the  ratio 
f/e'  =  —  A/2/014  is  finite  by  hypothesis  (see  §  184). 

In  a  telescopic  system  the  three  magnification-ratios  X,  Y  and  Z, 
as  we  saw  in  §184,  are  all  constant;  let  us  denote  their  values  here 
by  the  special  symbols  X0,  F0  and  Z0,  respectively.  So  soon  as  we 
know  the  values  of  any  two  of  these  magnitudes,  the  system  will  be 
completely  determined.  In  fact,  since  we  know  already  the  value  of 
the  finite  ratio  of  the  Focal  Lengths  of  the  Telescopic  System,  it  will 
be  sufficient  if  we  know  also  only  one  of  the  magnitudes  denoted  by 

XQ,    YQ,    £Q» 

The  diagram  (Fig.  96)  represents  the  case  of  the  combination  of 
two  non- telescopic  systems  into  a  telescopic  system,  which  is  the  case 


FIG.  96. 
TELESCOPIC  SYSTEM  RESULTING  FROM  THE  COMBINATION  OF  Two  NON-TELESCOPIC  SYSTEMS 

PLACED  TOGETHER  SO  THAT  A  =  0. 

now  under  consideration.  The  letters  in  this  figure  have  the  same 
meanings  as  they  have  in  the  preceding  figure,  so  that  they  do  not 
need  to  be  explained  again. 

The  Lateral  Magnification  is  evidently: 


If  we  wish  to  obtain  the  value  of  the  magnitude  F0  in  terms  of  the 
Focal  Lengths  of  the  component  systems,  we  have  from  the  definitions 
of  the  Focal  Lengths  (§  178): 


,  /,= 


A'V' 

**•  9  r   o 


tan  Z.A,F,V,' 


252  Geometrical  Optics,  Chapter  VII.  [  §  186. 

and  since  A  =  o,  so  that  the  points  E(  and  F2  are  coincident,  it  follows 
that 


and,  hence: 

r=F0  =  4. 

el 

Similarly,  the  Angular  Magnification  Z  =  Z0,  in  terms  of  the  Focal 
Lengths  of  the  component  systems,  may  be  obtained  as  follows: 
Consider  the  axial  points  Fl  and  E2,  which,  with  respect  to  the  com- 
pound system,  are  a  pair  of  conjugate  points;  evidently,  we  have: 

7  _  7  _  tan  ^A2E'2W2 
&  —  ^o  —   j. 

tan 

and  since 


tan  /.A2E2W2  =  — -, — ,     tan  ^Af^W^  =  — 7 — ,     A2W2  —  A^W^ 

62  Jl 

we  find: 

7—7    —  — 

^  ~  A)  -     '  • 

62 

And,  finally,  for  the  Axial  Magnification,  X  =  dx' /dx  =  x'/x  =  XQ, 
we  have,  since,  by  the  last  of  formulae  (120),  X  =  Y/Z, 

x  =  X0  =  77'  • 
Jiei 

2.  Combination  of  Telescopic  System  (7)  with  Non-Telescopic,  or 
Finite,  System  (77). 

Let  the  Telescopic  System  (I)  be  given  by  the  values  of  the  constant 
axial  and  lateral  magnification-ratios  Xv  F1?  respectively,  and  by  the 
positions  of  the  conjugate  axial  points  Mlt  M{  (Fig.  97).  Here,  as 
in  the  preceding  case,  we  assume  that  the  Principal  Axis  (x()  of  the 
Image-Space  of  system  (I)  and  the  Principal  Axis  (x2)  of  the  Object- 
Space  of  system  (II)  are  coincident.  If  F2  designates  the  position  of 
the  Focal  Point  of  the  Object-Space  of  system  (II),  the  relative 
positions  of  the  two  systems  may  be  assigned  by  the  value  of  the 
abscissa  of  the  point  F2  with  respect  to  the  point  M(.  Let  us  denote 
this  abscissa  by  the  symbol  a,  so  that  M[F2  =  a. 

A  ray  proceeding  parallel  to  the  Principal  Axis  fo)  of  the  Object- 
Space  of  system  (I)  will  also  be  parallel  to  the  Principal  Axis  (x{) 
of  the  Image-Space  of  this  system,  and,  emerging  finally  from  system 


§  186.] 


The  Geometrical  Theory  of  Optical  Imagery. 


253 


(II),  will  cross  the  Principal  Axis  (V)  of  the  Image-Space  of  the 
compound  system  at  the  Focal  Point  E'2  of  the  Image-Space  of  system 
(II),  which  is  likewise  also  the  Focal  Point  Ef  of  the  Image-Space  of 
the  compound  system.  And  since  the  position  of  E'2  is  given,  we  know, 
therefore,  the  position  of  E' '. 

The  position  of  the  other  Focal  Point  F  of  the  compound  system 
will  be  determined  if  we  ascertain  its  position  with  respect  to  the 


TxT— -^V 


FIG.  97. 

COMBINATION  OF  A  TELESCOPIC  WITH  A  NON-TELESCOPIC  SYSTEM  INTO  A  NON-TELESCOPIC 

COMPOUND  SYSTEM. 

given  axial  Object-Point  Mr  An  image-ray  which  emerges  from  the 
compound  system  in  a  direction  parallel  to  the  #'-axis  corresponds 
to  an  object-ray  which  crosses  the  x-axis  at  the  required  point  F,  and 
which  must  also  have  passed  through  the  Focal  Point  F2  of  the  Object- 
Space  of  system  (II).  Hence,  the  points  F  and  F2  must  be  a  pair  of 


. 


conjugate  axial  points  with  respect  to  the  telescopic  system  (I), 
since,  by  the  second  of  equations  (131), 


And 


we  obtain  immediately: 


M,F' 


whereby  the  position  of  the  Focal  Point  F  is  ascertained. 

It  only  remains  therefore  to  determine  the  magnitudes  of  the  Focal 
Lengths  /  and  er.     From  the  figure  we  obtain : 

A2X2  /2-tanZ; 


tan  Z  M{FQ 

where  Zl  denotes  the  constant  value  of  the  Angular  Magnification  of 
the  telescopic  system  (I).     Since  Zl  =  YlfXl,  we  have,  therefore: 


*i    ' 


254  Geometrical  Optics,  Chapter  VII.  [  §  186. 

where  /  is  determined  in  terms  of  the  known  constants  /2,  X±  and  Yt. 
Again, 

M(Q'      A2V2      4-tanZ^XF; 


tan 

and,  hence: 


IV 


whereby  the  Focal  Length  e'  of  the  Image-Space  of  the  compound 
system  is  determined  in  terms  of  the  given  constants  e'2  and  Fx. 

Since  the  magnitudes  denoted  by  Xlf  Ylt  f2  and  e'2  are  all  finite, 
it  is  evident  from  the  formulae  here  obtained  that  the  combination 
of  a  telescopic  system  with  a  non-telescopic  system  is  a  non-telescopic 
system.  If  the  system  (II)  were  the  telescopic  system,  the  procedure 
would  be  entirely  similar  to  that  given  above. 

3.  Both  Systems  Telescopic. 

Let  us  suppose  that  the  two  component  telescopic  systems  are  given 
by  the  values  of  their  constant  Axial  and  Lateral  Magnification-Ratios 
QZ~  ^a  , 


X. 

£- 

FIG.  98. 


COMBINATION  OF  Two  TELESCOPIC  SYSTEMS  INTO  A  TELESCOPIC  SYSTEM. 

Xlt  FT,  and  X2,  F2;  and  that  the  relative  position  of  the  two  systems 
is  given  by  the  positions  of  a  pair  of  conjugate  points  L,  L{  (Fig.  98) 
of  system  (I)  and  the  positions  of  a  pair  of  conjugate  axial  points 
M2,  M'  of  system  (II).  Let  us  write  a  =  L(M2. 

To  begin  with,  it  is  obvious  that  the  compound  system  is  also  tele- 
scopic.    Thus,  the  Lateral  Magnification  of  the  compound  system  is: 

L'Q      L'Q'    L{Q( 
=  ~LQ=L(Qi'~LQ'-    *VFa  =  a  constant; 

and  the  Axial  Magnification  of  the  compound  system  is: 

y  -  L'M'  -  L'M'    L*M2 

~~  LM  "=  T[M2  '  ~LM  :=Xi'X2  =  *  constant. 

Here  the  letters  M  and  L'  designate  the  positions  of  the  points,  which, 
with  respect  to  the  compound  system,  are  conjugate  to  the  given 


§  187.]  The  Geometrical  Theory  of  Optical  Imagery.  255 

points  M'  and  L,  respectively.     The  positions  of  the  points  M  and 
L'  may  be  determined  as  follows : 
Since  L{M2 :  LM  =  Xlt  and  L{M  =  a,  we  find: 


whereby  the  position  of  the  point  M  is  determined  relative  to  that 
of  the  given  point  L.     Again,  since  M' L':M2L(  =  X2,  we  have: 

M'L'  =  -a-A~2; 
whereby  the  point  L'  may  be  located  with  respect  to  the  given  point  M '. 

ART.  52.     GENERAL  FORMULA  FOR  THE  DETERMINATION  OF  THE  FOCAL 
POINTS  AND  FOCAL  LENGTHS  OF  A  COMPOUND  OPTICAL  SYSTEM. 

187.  We  shall  suppose  that  the  compound  system  consists  of  m 
component  systems,  and  we  shall  assume  that  the  Principal  Axis  (xk) 
of  the  Image-Space  of  the  kth  system  is  likewise  the  Principal  Axis  (xk+l) 
of  the  Object-Space  of  the  (k  +  i)th  system.  In  this  statement  the 
symbol  k  denotes  an  integer,  which  is  supposed  to  have  in  succession 
every  value  from  k=itok  =  m—  I. 

In  the  diagram  (Fig.  99),  Fk  and  E'k  designate  the  Focal  Points  of 
the  &th  system,  and  Ak  and  A'k  designate  the  Principal  Points  of  this 


Ak      £* 
Xk        dck" 


FIG.  99. 

DETERMINATION  OF  FOCAL  POINTS  (F,  £')  AND  FOCAL  LENGTHS  (/,  <?')  OF  A  COMPOUND 
OPTICAL  SYSTEM.  The  diagram  shows  the  Principal  Axes  (xk,  xk'),  the  Principal  Points  (Ak,  Ak') 
and  the  Focal  Points  (Fk,  Ek')  of  the  £th  member  of  the  system,  and  the  path  of  a  ray  traversing 

this  component. 

FkAk=fk.    Ek'Ak'  =  ek',    E*'Fk+i  =  ±k. 

FkMk'-\  =  xk,    Ek'Mk'=Xk',    AkJBk  =  hk  =  Ak'Sk'=:hk'. 
Z  Ak'-iMk'-\Bk'-i  =  <V-i,     Z  Ak'Mk'Bk''=  0*'. 

system.  To  an  axial  Object-Point  Ml  lying  on  the  Principal  Axis 
(tfj)  of  the  Object-Space  of  the  first  system  there  corresponds  an  axial 
Image-Point  M'm  lying  on  the  Principal  Axis  (x'm)  of  the  Image-Space 
of  the  last,  or  mth,  system.  A  ray  proceeding  originally  from  the 
point  Ml  will  cross  the  Principal  Axis  (xk)  of  the  Object-Space  of  the 
&th  system  at  the  point  M'k-i  and  the  Principal  Axis  (x'k)  of  the 


256  Geometrical  Optics,  Chapter  VII.  [  §  187. 

Image-Space  of  this  system  at  the  point  M'k.  Moreover,  let  Bk  and 
B'k  designate  the  points  where  this  ray  crosses  the  Principal  Planes  of 
the  kth  system,  and  let  us  write 


and,  also: 

Fk^k-i  = 

The  slope  of  the  ray  at  M'k-i  is 


The  Focal  Lengths  of  the  kth  system  will  be  denoted  by  /&,  ek  ;  thus, 


and,  finally,  the  interval  between  the  kth  and  the  (k  +  i)th  systems 
will  be  denoted  by 

A*  =  E'tFk+1. 

Evidently,  we  have  the  following  two  systems  of  equations: 

****  =  /*«*.       (&  =  i,  2,  •  -  •,  m),  (137) 

and 

-f-  Ai,     (k  =  i,  2,  •  •  -,  m  -  i).  (138) 


i.  Determination  of  the  Positions  of  the  Focal  Points  F,  Ef  of  the 
Compound  System. 

From  the  two  systems  of  equations  (137)  and  (138),  we  obtain  by 
process  of  successive  elimination: 


Now  if  the  Object-Point  Ml  coincides  with  the  Focal  Point  F  of  the 
Object-Space  of  the  compound  system,  the  Image-Point  M'm  will  be 
the  infinitely  distant  point  of  the  Principal  Axis  (#')  of  the  Image- 
Space  of  the  compound  system;  and  in  this  case: 

*!  =  F,F,     xm  =  oo. 

Introducing  these  values,  we  obtain  the  abscissa  of  the  Focal  Point 
F  of  the  compound  system  in  the  form  of  a  continued  fraction  as 
follows: 


§  187.] 


The  Geometrical  Theory  of  Optical  Imagery. 


257 


fm-\em-l 


(139) 


On  the  other  hand,  if  we  suppose  that  the  Image-Point  M'm  coincides 
with  the  Focal  Point  E!  of  the  Image-Space  of  the  compound  system, 
the  Object-Point  Ml  will  be  the  infinitely  distant  point  of  the  Principal 
Axis  (x)  of  the  Object-Space  of  the  compound  system;  so  that  in 
this  case  we  shall  have: 

x'm  =  E'mE'; 


=00 


and,  by  a  process  precisely  analogous  to  the  above,  we  obtain: 

/«X» 

~  ~~~  -f       '  * 

A  I  Jm—l^m—l 


(140) 


whereby  the  abscissa  of  the  Focal  Point  E'  of  the  compound  system 
with  respect  to  the  Focal  Point  E'm  of  the  mth  system  is  expressed 
also  in  the  form  of  a  continued  fraction. 

The  continued  fractions  which  form  the  right-hand  sides  of  equations 
(139)  and  (140)  may  be  expressed  in  the  form  of  determinants.  Thus, 
writing 

A  A' 

F,F  =  -g     and     E'Jg  =  -g , 

we  have  for  the  numerators  A,  A'  and  the  denominators  B,  B'  the 
following  determinant-arrays : 


fl 

0 

o 

o 

o 

0 

0 

o 

-e{ 

0 

0 

o 

0 

0 

o 

fi 

A2 

-4 

o 

o 

o 

0 

0 

/3 

AS 

-  e± 

0 

0 

0 

0 

O 

0 

o 

/*., 

A™-, 

18 


258 


Geometrical  Optics,  Chapter  VII. 


§187. 


B'  = 


fm 

0 

0 

o 

o 

0 

0 

O 

-4 

o 

0 

o 

o 

o 

0 

fro-l 

Am_2 

-4-2 

o 

0 

o 

0 

0 

L-2 

A.-3 

-4-3  ••• 

0 

0 

. 

• 

• 

-4 

0 

0 

0 

o 

o 

/a 

A! 

I 

0 

0 

0 

o 

0 

O 

0 

A! 

—  4 

o 

o 

o 

0 

o 

/2 

A2 

-4 

o 

0 

o 

0 

0 

/3 

A3 

-  <?4 

0 

0 

0 

o 

0 

0 

o 

/„-! 

"I7_; 

I 

0 

o 

0 

o 

o 

o 

o 

AOT_i 

-4-i 

0 

o 

o 

0 

0 

/*-! 

V-i 

-4-2 

o 

o 

o 

o 

O 

L-2 

A,_3 

-c,  ••• 

0 

0 

. 

• 

•   . 

. 

-4 

o 

O 

0 

0 

o 

/J 

A! 

A  mere  inspection  of  the  last  two  arrays  will  show  that  the  denomi- 
nators B  and  B'  are  equal. 

2.  Determination  of  the  Focal  Lengths  /,  e'  of  the  Compound  System. 

In  order  to  determine  the  Focal  Length  e'  of  the  Image-Space  of 
the  Compound  System,  let  us  consider  an  object-ray  parallel  to  the 
Principal  Axis  (x)  of  the  Object-Space,  to  which  corresponds,  therefore, 
an  image-ray  which  goes  through  the  Focal  Point  E'  of  the  Image- 
Space.  Accordingly,  for  this  ray: 

*i=<*>,     xm  =  E'mE'- 
By  the  definition  of  the  Focal  Length  e',  we  have: 


187.] 


The  Geometrical  Theory  of  Optical  Imagery. 


259 


e' 

where  Q'm  =  Z.A'mE'B'm. 

This  equation  may  be  written: 


tan 01  ' 


ef 


i       tan  0{    tan  0'2        tan  0^_j 


tan  0[    tan  d'2    tan  6$          tan  B'm  ' 
and  since  e{  =  hj tan  O'lt  and  since,  also,  by  formula  (119): 

tan0;  *& 

^      — "  > 

tan  0A_j  e^ 

we  have: 


*»•••*« 


(141) 


Putting  xl  =  oo,  we  obtain  from  the  two  systems  of  equations  (137) 
and  (138): 


where  Rfc  is  used  to  denote  the  &th  term  of  this  product  of  continued 
fractions.     Writing 

R  --* 

"~ 


we  may  express  each  of  these  continued  fractions  as  the  quotient  of 
two  determinants  as  follows: 


f 

\     ~~  g* 

0                   0 

o 

o 

0 

/.       V. 

f 

0 

...       o 

0 

f 

°                 /*-! 

o 

o 

o 

O              O 

fk-2              \-3 

-4-3 

o 

0 

. 

• 

• 

o 

-e', 

0            0 

0                 0 

o 

...   /2 

A, 

260 


Geometrical  Optics,  Chapter  VII. 


[  §  187. 


/*-. 

o 


f 


k-2 


0 
0 

0 

...       o 
...       o 
...       0 

0 
0 

o 

-4 

000 

A  mere  inspection  of  these  two  determinants  shows  that  we  have 


accordingly, 


since  we  must  put  Ql  =  i.     Thus,  we  find: 


e,  -e. 


TO-1 


(I42) 


Again,  in  order  to  determine  the  Focal  Length  /  of  the  Object-Space 
of  the  compound  system,  consider  an  image-ray  parallel  to  the  Prin- 
cipal Axis  (#')  of  the  Image-Space;  to  which  corresponds  an  object- 
ray  which  goes  through  the  Focal  Point  F  of  the  Object-Space. 
Accordingly, 


x   = 


xm  =  oo. 


By  the  definition  of  the  Focal  Length  /,  we  have  : 


tan 


where  Q\  = 


and  since 


^    This  equation  may  be  written  : 


tan  0      tan  6'        tan  0- 


h'm 


tan  0,     tan  0,         tan  dm_2    tan  0OT_ 


h'm 


=  /„ 


tan  0m_i 
and  since,  also,  by  formula  (119),  we  have: 

tan  01  fk 

tan  0&—1  Xk ' 


§  187.]  The  Geometrical  Theory  of  Optical  Imagery. 

we  obtain  here: 

/-(-i)-1 4^r^^~- 


261 


(143) 


Putting  ^  =  oo,  we  derive  from  the  two  systems  of  equations  (137) 
and  (138): 


•;•*;••.*'  ,  =  (A 


A2+r 


m-im- 

-2 


where  1^  is  used  to  denote  the  &th  term  of  this  product  of  continued 
fractions.     Writing 


we  may  express  each  of  these  continued  fractions  as  the  quotient  of 
two  determinants  as  follows: 


A* 

/ 
—  .£fc+l           O                 O 

o 

o 

o 

/HI 

A4+I  -e;+2    o 

o 

o 

o 

O 

/&+2         Afr+2    —  tffc+a 

o 

o 

o 

O 

O            /fc+3        Afc+« 

t    •  •  •        o 

o 

. 

. 

. 

o 

-4-i 

0 

O                O                O 

0 

Jm—  I 

Am_! 

A*+I 

—  4+2           0                 0 

0 

o 

o 

/*+2 

A*+2  -4+3    o 

o 

o 

0 

0 

4+3       A&+3    -el+4 

0 

o 

0 

. 

. 

. 

o 

-«Li 

o 

o           o            o 

0 

'm-l 

Am_t 

Here  also  it  is  evident  that: 

Pi 

accordingly, 


262  Geometrical  Optics,  Chapter  VII.  [  §  187. 

Pi  , 


Xl'X2  '  '  '  Xm—i 


since  we  must  put  Q'm-i  =  I.     Thus,  we  find: 


Comparing  the  two  determinant-arrays  denoted  by   Pk  and   Pi, 
we  see  that  Pw_!  and  P(  are  equal;  if,  therefore,  we  write: 


formulae  (142)  and  (144)  may  be  written: 

e'  =  6l'e2D'e->    /=(_I)-i^A_J^  (I45) 

Consequently,  also: 

4  =  (-  ^m-Ji'f^  •  ' L ^  ^     6) 


If,  therefore,  we  know  the  determination-constants  and  the  rela- 
tive positions  of  the  members  of  the  compound  system,  the  formulae 
which  we  have  here  obtained  will  enable  us  to  determine  the  Focal 
Points  and  the  Focal  Lengths  of  the  compound  system.1 

1  In  regard  to  the  literature  dealing  with  the  subject  of  Art.  51,  the  following  is  a  partial 
list  of  the  writers: 

With  reference  to  the  matters  treated  in  \\  185  and  186,  consult:  S.  CZAPSKI:  Theorie 
der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  pages  46-51.  Also,  with  reference 
to  \  187,  see  CZAPSKI,  pages  51-53.  See  also  E.  WANDERSLEB:  Die  geometrische  Theorie 
der  optischen  Abbildung  nach  E.  ABBE,  Chapter  III  of  Vol.  I  of  Die  Theorie  der  optischen 
Instrumente  (Berlin,  1904),  edited  by  M.  VON  ROHR;  pages  112-121.  Also,  Dr.  J.  CLAS- 
SEN'S Mathematische  Optik  (Leipzig,  1901),  Art.  46. 

With  special  reference  to  3  187: 

A.  F.  MOEBIUS:  Beitraege  zu  der  Lehre  von  der  Kettenbruechen,  nebst  einem  Anhang 
dioptrischen  Inhalts:  CRELLES  Journ.,  vi.  (1830),  215-243. 

F.  W.  BESSEL:  Ueber  die  Grundformeln  der  Dioptrik:  Astr.  Nach.,  xviii.  (1841), 
No.  415,  pages  97-108. 

F.  LIPPICH:  Fundamentalpuncte  eines  Systemes  centrirter  brechender  Kugelflaechen: 
Mitt,  naturw.  Ver.  Steiermark,  ii.  (1871),  429-459. 

S.  GUENTHER:  Darstellung  der  N aeherungswerte  von  Kettenbruechen  in  independenter 
Form:  Habil.-Schr.  Erlangen,  1873. 

O.  ROETHIG:  Die  Probleme  der  Brechung  und  Reflexion  (Leipzig,   1876). 

F.  MONOYER:  Theorie  generate  des  systemes  dioptriques  centres:  Paris,  Soc.  Phys. 
Seances,  1883,  148-174. 

L.  MATTHIESSEN:  Allgemeine  Formeln  zur  Bestimmung  der  Cardinalpunkte  eines 
brechenden  Systems  centrierter  sphaerischer  Flaechen;  mittels  Kettenbruchdetermin- 
anten  dargestellt:  Zft.  f.  Math.  u.  Phys.,  xxix.  (1884),  343-350. 


CHAPTER  VIII. 

IDEAL    IMAGERY   BY   PARAXIAL    RAYS.      LENSES   AND    LENS-SYSTEMS. 
ART.  53.     INTRODUCTION. 

188.  The  geometrical  theory  of  optical  imagery,  as  developed  in 
the  preceding  chapter,  is  entirely  independent  of  the  physical  laws  of 
Optics.  The  fundamental  and  single  assumption  on  which  the  theory 
rests  is  that  of  point-to-point  correspondence,  by  means  of  rectilinear 
rays,  between  Object-Space  and  Image-Space.  With  regard  to  the 
angular  apertures  of  the  bundles  of  rays  employed  in  the  production 
of  the  image,  as  also  with  regard  to  the  dimensions  of  the  object  to  be 
portrayed,  absolutely  no  conditions  were  imposed.  In  that  chapter 
we  were  not  at  all  concerned  with  the  mechanism  whereby  an  image 
may  be  realized;  we  merely  assumed  that  such  imagery  was  possible 
and  investigated  the  laws  thereof.  Whatever  practical  difficulties  may 
lie  in  the  way  of  realizing  the  geometrical  condition  of  collinear  cor- 
respondence, we  have  not  yet  encountered  them,  as  we  shall  have  to 
do  hereafter. 

The  investigation  of  the  Refraction  of  Paraxial  Rays  of  monochro- 
matic light  through  a  centered  system  of  spherical  refracting  surfaces 
had  prepared  the  way  for  the  geometrical  theory  of  optical  imagery; 
for  in  this  special,  and,  to  be  sure,  more  or  less  impractical,  case  we 
saw  that  there  was  strict  collinear  correspondence  between  Object- 
Space  and  Image-Space.  Hence,  here  at  any  rate,  the  formulae  of  the 
preceding  chapter  are  immediately  applicable.  The  theory  of  the  re- 
fraction of  paraxial  rays  through  a  centered  system  of  lenses  was  first 
fully  worked  out  by  GAUSS1;  and,  hence,  the  imagery  which  we  have 
under  these  circumstances  is  frequently  called  "GAiissian  Imagery". 

The  determination-data  of  a  centered  system  of  spherical  surfaces 
are  usually  the  refractive  indices  of  the  successive  isotropic  media, 
the  radii  of  the  spherical  surfaces,  and  the  distances  between  the  con- 
secutive vertices.  If  we  introduce  these  constants  into  the  general 
formulae  of  the  preceding  chapter,  we  shall  obtain  not  only  all  the  re- 
sults which  for  the  case  of  Paraxial  Rays  we  have  previously  obtained 
by  independent  methods,  but  also  a  number  of  new  and  useful  formulae, 

1  C.  F.  GAUSS:  Dioptrische  Untersuchungen  (Goettingen,  1841).  See  also  paper  by 
F.  W.  BESSEL,  entitled  "  Ueber  die  Grundformeln  der  Dioptrik  "  (Astr.  Nach.,  xviii.,  1841, 
No.  415,  pages  97-108). 

263 


264  Geometrical  Optics,  Chapter  VIII.  [  §  190. 

particularly  in  regard  to  the  Focal  Lengths  and  the  Magnification- 
Ratios  of  the  optical  system. 

189.  In  the  case  of  a  Single  Spherical  Surface,  of  radius  r,  separating 
two  media  of  refractive  indices  n,  n',  we  found  (§  124)  that  the  focal 
lengths  /,  e'  were  given  by  the  following  formulae  : 


If  the  vertex  of  the  surface  is  at  the  point  designated  by  A,  and  if 
the  positions  of  the  focal  points  are  designated  by  F,  E',  then/  =  FAy 
e'  =  E'A\  and,  consequently  (§180),  the  Principal  Points  coincide 
with  each  other  at  the  vertex  A  .  The  Nodal  Points,  evidently,  coin- 
cide at  the  centre  C. 

If,  therefore,  M,  M'  designate  the  positions  of  a  pair  of  conjugate 
axial  points  of  the  Spherical  Surface,  and  if,  according  to  our  previous 
system  of  notation,  we  put  AM  =  «,  AM'  =  u',  we  obtain  at  once 
from  formulae  (128)  : 


V  =  —  =. 

y  e'u 

_  tanfl'  _  u_ 
~  tan0  ~  w' 


i  =o, 
fu'      n 


(148) 


The  first  two  of  these  formulae  will  be  recognized  as  identical  with 
formulae  (85)  and  (86). 

ART.  54.      THE  FOCAL  LENGTHS  OF  A  CENTERED  SYSTEM  OF  SPHERICAL 

SURFACES. 

190.  In  §  137  we  showed  how  to  determine  the  positions  of  the 
Focal  Points  F  and  E'  of  a  centered  system  of  spherical  refracting 
surfaces.  Thus,  using  the  same  system  of  notation  as  was  employed 
there,  we  obtained  two  sets  of  equations  of  the  following  types: 


'/    /  /  '\         '    /    /  /    x 

nk(i/rk  -  i/O  =  fi^d/fji  -  i/O 


1 

;  J 


wherein  we  must  give  k  in  succession  all  integral  values  from  k  =  i  to 
k  =  m\  noting  also  that  d0  =  o.  The  diagram  (Fig.  100)  shows  the 
path  BkBk+l  of  a  ray  between  the  &th  and  the  (k  +  i)th  surfaces  of 


§  190.]  Ideal  Imagery  by  Paraxial  Rays. 

the  centered  system  of  spherical  surfaces;  and  we  have: 


265 


Also,  in  accordance  with  our  previous  notation,  let  us  write : 

=   Jk.         Z    4 


Here  Ak  and  Ck  designate  the  vertex  and  centre  of  the  kth  spherical 
surface,  and  Mk  designates  the 
point    where    the    paraxial  ray 
crosses  the  axis  after  refraction 

at  the  kth  surface.  x 

In  order  to  determine  the 
Focal  Length  er  of  the  Image- 
Space  of  the  system  of  m  spher-  FlG- 10°- 

i  r  r     ,  i  PATH  OF  A  PARAXIAL  RAY  BETWEEN  £th  AND 

ICal    Surfaces,    in    terms    of     the       u  +  1)th  SURFACES  OF  A  CENTERED  SYSTEM  OF 

magnitudes  denoted  by  r,  n  and     SPHERICAL  SURFACES. 

d,  let  us  consider  a  ray  which  in         AkMk' =  **'•  AMMk> =  »*«•  A^M  =  dk, 

the  Object-Space  is  parallel   to  A"c"  =  ^  **»  =  **•  A^Bk^hk^ 

.  .  £.  AkMkBk  —  9k. 

the  optical  axis  (^  =  oo),  and 

which,  therefore,  in  the  Image-Space  crosses  the  optical  axis  at  the 
Focal  Point  E'(um  =  AmE').  By  the  definition  of  the  Focal  Length 
ef  given  in  §  178,  we  have: 


where  0'm  =  Z.AmE'Bm.     Thus,  we  may  write: 

h2    h3          hm      tan  B'M 
By  the  diagram  we  have  obviously : 

T?*-— ^-,    tan«;---f; 

rlk+i       Uk+\  Uk 

and,  accordingly,  we  derive  the  following  formula: 

e  =  E'Am  -  - 


(ISO) 


where  the  magnitudes  uk,  u'k  can  be  determined  in  terms  of  the  known 
constants  by  means  of  the  (2m  —  i)  equations  (149). 


266  Geometrical  Optics,  Chapter  VIII.  [  §  193. 

191.  Formulae  (149)  and  (150)  may  be  written  in  the  following 
forms,  adapted  to  the  logarithmic  computation  of  the  path  of  a  paraxial 
ray  through  a  centered  system  of  spherical  surfaces: 

i         i      w/,_i       i     nk  —  nk-\ 

AM  -  A I  _  /W  _  -V  _  'W.  _ 


nk        rk 

_L         i 

u'k    i  -  dk/uk  ' 


um 


(ISO 


In  Chapter  X  there  is  given  a  numerical  illustration  of  the  use  of 
formulae  (151)  in  calculating  the  value  of  um  =  A'mEf.  When  the  focal 
length  e'  of  the  system  is  given,  the  last  of  these  formulae  gives  a  very 
convenient  way  of  checking  the  logarithmic  computation. 

192.  The  Focal  Length  f  of  the  Object-Space  may  be  determined  in 
an  analogous  way  by  considering  a  second  ray  for  which  u±  =  A1F, 
um  =  oo  .     Thus,  we  shall  obtain  a  similar  formula  for  /,  as  follows  : 

u^^ 

J  1      a  i      .  a  l      .    .    .  a  i  \      J       ' 

Ul    U2         Um_l 

in  which,  however,  the  magnitudes  denoted  by  ukl  uk  will,  of  course, 
not  have  the  same  values  as  they  have  in  formula  (150),  because  here 
they  have  reference  to  a  different  ray. 

193.  Ratio  of  the  Focal  Lengths  /  and  e'. 

The  system  of  spherical  surfaces  may  be  regarded  as  a  compound 
system  formed  by  the  combination  of  m  spherical  surfaces;  thus,  if 
/&,  ek  denote  the  Focal  Lengths  of  the  kth  spherical  surface,  we  must 
have,  according  to  the  third  of  formulae  (147)  : 


hence,  applying  formula  (146)  at  the  end  of  Chapter  VII,  we  obtain 
directly  : 


n 


In  case,  therefore,  all  of  the  spherical  surfaces  are  refracting  surfaces, 
we  obtain  the  following  formula  : 


§  194.[  Ideal  Imagery  by  Paraxial  Rays.  267 

but  if  the  system  consists  of  both  refracting  and  reflecting  surfaces, 
and  if  the  number  of  the  reflecting  surfaces  is  odd,  the  formula  will  be  : 

93  ?=+?'  <'53«) 

nm 

In  connection  with  these  formulae,  it  is  well  to  remind  the  reader 
again,  as  was  stated  in  §  176,  that  in  the  case  of  a  centered  system  of 
spherical  surfaces,  the  positive  direction  along  the  optical  axis  is  deter- 
mined by  the  direction  of  the  incident  axial  ray;  and  no  matter  if  the 
direction  of  this  ray  should  be  reversed  by  one  or  more  reflexions,  the 
positive  direction  of  the  optical  axis  remains  unchanged.  Thus,  the 
Focal  Lengths,  Radii,  etc.,  are  to  be  reckoned  positive  or  negative, 
according  as  they  are  measured  in  the  same  direction  as,  or  in  the 
opposite  direction  to,  the  direction  of  the  incident  axial  ray  (see  §  26). 

The  useful  result,  which  we  have  just  found,  may  be  stated  as  follows  : 

In  any  centered  system  of  spherical  surfaces,  the  absolute  value  of  the 
ratio  of  the  two  focal  lengths  is  equal  to  the  ratio  of  the  indices  of  refraction 
of  the  first  and  last  media.  This  ratio  is  negative,  except  in  the  case 
when  an  odd  number  of  the  m  spherical  surfaces  are  reflecting.  In 
this  exceptional  case  the  ratio  fje'  is  positive. 

In  particular,  if  the  media  of  the  Object-Space  and  Image-Space 
are  identical  in  substance,  that  is,  if  nv  =  nm,  the  absolute  values 
of  the  focal  lengths  are  equal.  In  this  case,  which  is  so  often  realized 
in  optical  instruments,  if  we  suppose  that  all  the  surfaces  are  refract- 
ing, or  that  an  even  number  of  them  are  reflecting,  we  shall  have 
/  =  —  e',  and,  therefore,  the  nodal  points  N,  N'  will  coincide  with 
the  principal  points  A,  A',  respectively;  for,  according  to  §  180,  we 
have: 

=  A'E'=-e'=f=FA,    and    E'N'  =  AF  =  -/  =  e'  =  E'A'. 


ART.  55.      SEVERAL  IMPORTANT  FORMULAE   FOR  THE   CASE   OF  THE   RE- 

FRACTION OF  PARAXIAL  RAYS  THROUGH  A  CENTERED  SYSTEM 

OF  SPHERICAL  SURFACES. 

194.     Robert  Smith's  Law. 

According  to  the  second  of  formulae  (120),  we  found  that  in  an  optical 
system  the  product  of  the  Lateral  Magnification  Y  =  y'/y  and  the 
Angular  Magnification  Z  =  tan  0'/tan  0  is  constant;  that  is, 

Y-Z=  -f/e', 
or 

y'-tanfl'        _  / 

y  -  tan  0  e  ' 


268  .  Geometrical  Optics,  Chapter  VIII.  [  §  194. 

If  the  indices  of  refraction  of  the  Object-Space  and  Image-Space  are 
denoted  by  n  and  n',  respectively,  formulae  (153)  and  (1530)  may  be 
written  as  follows: 

/_    » 

e  ~  '  *  n" 

and  combining  this  with  the  preceding  equation,  we  obtain  one  of  the 
most  important  relations  of  Geometrical  Optics,  as  follows: 

w'y'tan  tf  =  ±  ny  tan  6,  (154) 

or 

n'y'  e'  =  ±  ny0, 


where  the  lower,  or  negative,  sign  applies  only  in  case  we  have  an  odd 
number  of  reflexions. 

In  most  German  books  on  Optics  this  formula  is  called  the  "LA- 
GRANGE-HELMHOLTZ"  Equation.  A  very  interesting  account  of  the 
history  of  this  celebrated  law  of  Optics  is  given  in  a  note  at  the  end  of 
P.  CULMANN'S  article  on  "Die  Realisierung  der  optischen  Abbildung", 
which  is  Chapter  IV  of  the  first  volume  of  Die  Theorie  der  optischen 
Instrumente,  edited  by  M.  VON  ROHR  (Berlin,  1904).  CULMANN  con- 
cludes that  the  formula  should  be  called  the  "HELMHOLTZ  Equation", 
inasmuch  as  HELMHOLTZ*  gave  the  equation  in  the  form  in  which  it 
is  now  used.  He  suggests  also  that  it  might  be  called  the  "SMITH- 
HELMHOLTZ"  Equation.  HELMHOLTZ  himself  attributed  the  law  to 
LAGRANGE,  who  published,  in  1803,  a  special  case  of  the  law.2  But 
Lord  RAYLEIGH3  has  pointed  out  that  ROBERT  SMITH,4  with  whose 
work  LAGRANGE  was  undoubtedly  acquainted,  had  enunciated  the 
law  for  the  special  case  of  a  system  of  infinitely  thin  lenses  as  early  as 
1738,  deducing  it  from  COTES'S  Theorem  (see  Art.  42,  especially  §  152). 
SMITH  treats  the  whole  subject  in  a  very  masterful  way,  and  recog- 
nizes fully  the  importance  and  the  consequences  of  the  law,  although 
it  is  true  he  did  not  establish  it  for  the  most  general  case.  The  for- 
mula, as  applied  to  a  centered  system  of  spherical  surfaces,  was  given 
by  LUDWIG  SEIDEL,  in  1856,  in  his  paper  "Zur  Dioptrik,"  etc.,  pub- 
lished in  Vol.  xliii.,  No.  1027,  of  Astronomische  Nachrichten  (pages 
290-303)  :  see  formulae  (8)  in  SEIDEL'S  paper. 

XH.  HELMHOLTZ:   Handbuch  der  physiologischen  Optik  (1867),  p.  50. 

2  J.  L.  DE  LAGRANGE:  Sur  une  loi  generate  d'optique  :  Memoires  de  V  Academic  de 
Berlin  (1803),  3-12. 

8J.  W.  STRUTT,  Lord  RAYLEIGH:  itotes,  chiefly  Historical,  on  some  Fundamental 
Propositions  in  Optics:  Phil.  Mag.  (5),  xxi.  (1886),  pp.  466-476. 

4  ROBERT  SMITH:  A  Compleat  System  of  Opticks  (Cambridge,  1738);  Book  II,  Chap.  V. 


§  195.]  Ideal  Imagery  by  Paraxial  Rays.  269 

195.  Formulae  of  L.Seidel.  The  following  formulae,  due  to  L.SsiDEL,1 
will  be  frequently  employed  in  the  Theory  of  Spherical  Aberration. 

Let  Mlt  Ml  designate  the  positions  of  two  points  on  the  optical 
axis  of  a  centered  system  of  spherical  refracting  surfaces,  and  let 
AlMl  =  ult  AlMl  =  ulf  where  Al  designates  the  vertex  of  the  first 
surface.  Consider  two  Paraxial  Rays,  which,  before  refraction  at  the 
first  surface,  cross  the  optical  axis  at  Mlt  M^  and  which,  before  re- 
fraction at  the  kth  surface,  cross  the  optical  axis  at  the  points  desig- 
nated by  JlfjLi,  AfJLi,  and  which  are  incident  on  the  kth  surface 
at  points  designated  by  Bk,  Bk,  respectively.  In  agreement  with  our 
previous  notation,  we  shall  write: 

4*Mfc-i  =  uk,     AkMk  =  u'k,     AkBk  =  hi 
Since,  by  formulae  (149), 


we  obtain: 

Also,  since 
and 

we  find: 


and  combining  the  two  equations  thus  obtained,  we  derive  the  first 
of  SEIDEL'S  formulae,  as  follows: 


'L.  SEIDEL:   Zur  Dioptrik  :    Astr.  Nachr.,  xxxvii.  (1853).  No.  871,  pages  105-120. 


270  Geometrical  Optics,  Chapter  VIII.  [  §  195. 

=  nk_  A-  A-i 


Wi/Zi/li  I        •   —  I. 

\«1          »!/ 


If  we  introduce  here  the  so-called  "zero-invariant"  (see  §126),  and 
write 

n(i/r  -  i/u)  =  n'(i/r  -  i/u')  =  J, 

n(i/r  -  i  /a)  =  «'(i/r  -  i/u')  =  J, 
the  formulae  above  may  be  written  as  follows  : 


Moreover,  as  can  be  seen  easily  from  Fig.  100,  we  have: 


hk_l          uk_i  '         h  Jf_1  «^_!  ' 

and,  hence,  combining  these  equations,  and  using  formula  (155),  we 
obtain: 


.  fj  w     N     -i  __  T.  h  ,j        w^ 

JL~     ~  L       ~  \Jk-i  ~~  Jk-\)    '      —  »if*iv««i      *v!v 
%-i     "i-i  %-i  % 

or 


k-i 


Giving  k  in  succession  the  values  2,  3,  •  •  •  k,  and  adding  all  the 
equations  thus  obtained,  we  derive  the  second  of  SEIDEL'S  Formulae, 
as  follows: 

Moreover,  since 
and 

we  can  write: 


.,      (     I  I     \  ^1^  fX 

,nk  I    —7 r  I  =      /  7     (  JT  —  JJ, 

1  *\uA     wj     wAv  ' 


§  196-]  Ideal  Imagery  by  Paraxial  Rays.  271 

or 

hJh  _  !M»  _  ^A  /  T  _ .  F  \    M*i  (kk    hk 

'  '       ~~      '  7i     ^     1  —       1'    ~  '      I    zT~  —   JL~ 

and,  hence,  by  (156),  we  derive  also  the  following: 

^  -^J^^-y };    (X57) 

which  is  the  third  of  SEIDEL'S  Formulse. 
The  expression 

^—rf   —'         7.  rT  > 


which  occurs  frequently,  especially  in  SEIDEL'S  optical  formulae,  may 
be  transformed  as  follows: 


=  _  2_  ..L    _ 

/^i  V^i    w;y  ^  HI  \n{  n'J^     r 

j4          u,        ^  i  (  uk    _u'k\ 

nX    JhU^cSAiVJC    %y 

=  j^L  ___^L  i  y  M*M*  (^    T    __i_ 
»X    ^1^5    *=i  *J  \»i-X    »; 

and,  finally: 


_  . 

n'^-h^-h,      nkhzk      nft  hi    rk 

where  A  is  the  symbol  of  operation  introduced  in  §  126;  so  that  here 


196.  In  order  to  illustrate  some  of  the  uses  of  these  formulae,  we 
shall  apply  them,  as  SEIDEL  does,  to  determine  the  positions  of  the 
Focal  Points  F,  Ef  and  of  the  Principal  Points  A,  A'  and  the  magni- 
tudes of  the  Focal  Lengths  /  =  FA  ,  e'  —  E'A  '  of  a  centered  system  of 
m  spherical  refracting  surfaces. 

We  shall  suppose  that  we  know  completely  the  path  of  a  certain 
paraxial  ray  through  the  system,  for  example,  the  ray  which  in  the 
object-space  crosses  the  optical  axis  at  the  point  designated  by  M^ 
and  which  meets  the  first  spherical  surface  at  the  point  designated  by 


272  Geometrical  Optics,  Chapter  VIII.  [  §  196. 

Br  Thus,  instead  of  the  usual  determination-constants  of  the  optical 
system,  viz.,  the  radii  (r)  and  the  axial  thicknesses  (d),  we  shall  suppose 
that  we  have  given  here  the  elements  of  this  ray,  so  that  the  system 
is  determined  by  the  intercepts  (u)  of  the  ray  on  the  optical  axis 
and  by  the  incidence-heights  (ti);  as  is  perfectly  possible,  since  the 
former  magnitudes  can  be  expressed  in  terms  of  the  latter,  as  follows : 


0), 


dk  =  uk  - 


In  deriving  the  following  formulae,  we  shall,  chiefly  for  the  sake  of 
brevity  in  writing,  introduce  a  constant  of  the  optical  system,  denoted 
by  the  symbol  F,  and  defined  by  the  expression: 


,-. 

?       n'nhl  n'^-h' 

which  may  be  written  also  as  follows: 

i         M«*f*«X  l  \(l\ 

F=  -^STTT  v» 

as  is  evident  from  the  transformation  given  in  formula  (158). 

The  positions  of  the  Focal  Points  F,  JL'  can  be  found  by  means  of 
formula  (157).  Thus,  if  in  this  formula  we  put,  1st,  ut  =  A1F,  um  —  oo, 
and,  2nd,  ux  =  oo,  u'm  =  AmE',  we  obtain 


(160) 


In  order  to  determine  the  positions  of  the  Principal  Points  A,  A' 
of  the  centered  system  of  spherical  surfaces,  we  recall  (§139)  that  this 
pair  of  conjugate  points  is  characterized  by  the  condition: 

n'm  k=l  **k 

where  ul  =  A^,  um  =  AmA'\  which,  since 

«          f 

fcl'ttT' 


k 


§  197.[  Lenses  and  Lens-Systems.  273 

may  be  expressed  by  the  following  relation  : 


In  formula  (157),  therefore,  we  must,  1st,  substitute  n-Ji^jn^u-^  for 
hju'm,  solve  for  ulf  and  put  u^  =  AvA  ;  and,  2nd,  substitute  n^hjn^ 
for  A!/!*!,  eliminate  (/x  —  JJ  by  (155),  solve  for  u^,  and,  finally,  put 
u'm—  AmA'.  Performing  both  of  these  ope  rations,  and  making  use  of 
formula  (159)  each  time,  we  obtain  the  following  formulae: 


(161) 


Having  determined  the  positions  of  the  Focal  Points  and  of  the 
Principal  Points,  we  can  find  at  once  the  magnitudes  of  the  Focal 
Lengths/,  e' \  for 

f=FA  =  FAl  +  A,A, 

e'  =  E'A'  =  E'Am  +  AmAf, 
and,  hence,  by  formulae  (160)  and  (161): 

/  =  F/n'»,     e'  =  -  F/n,.  (162) 

If  the  given  ray,  to  which  the  symbols  u,  h  refer,  is  incident  on  the 
first  surface  of  the  system  in  a  direction  parallel  to  the  optical  axis, 
we  must  put  u^  =  °o  in  the  above  formulae,  whereby  we  obtain  for 
the  Focal  Lengths: 

^ y^.-.ic    ,__«;•«;•••.«:        (I6) 

nm    u2-u,---um'  u2-u3'"Um 

in  agreement  with  the  results  as  expressed  by  formulae  (150)  and  (153). 

LENSES  AND  LENS-SYSTEMS. 
ART.  56.     THICK  LENSES. 

197.  A  centered  system  of  two  spherical  refracting  surfaces  (m  =  2) 
is  called  a  Lens.  In  the  following  discussion  of  Lenses  it  will  be  as- 
sumed that  the  media  of  the  incident  and  emergent  rays  are  identical 
in  substance,  and  the  symbol 

n  =  n'Jn,  =  n'Jn'2 

19 


274  Geometrical  Optics,  Chapter  VIII.  [  §  197 

* 

will  be  employed  to  denote  the  relative  index  of  refraction  of  the 
medium  of  the  lens-substance  and  the  surrounding  medium.  A  Lens 
is  usually  described  by  assigning:  (i)  The  magnitude  of  the  relative 
index  of  refraction  («);  (2)  The  positions  Alt  A2  of  the  vertices  of 
the  two  spherical  surfaces,  whereby  the  optical  axis  of  the  lens  may 
be  directly  determined,  both  as  to  its  position  and  as  to  its  direction, 
by  drawing  the  straight  line  from  Al  to  A2.  The  distance 

AiA2  =  d, 

called  the  "thickness"  of  the  Lens,  is  reckoned  always  in  the  positive 
direction  of  the  optical  axis,  so  that  d  is  essentially  a  positive  magni- 
tude; and  (3)  The  magnitudes  and  signs  of  the  radii  rl  =  A^C^ 
r2  =  A2C2.  Employing  the  same  letters  and  notation  as  are  used  in 
the  preceding  portion  of  this  chapter,  we  shall  regard  a  Lens  as  a  com- 
pound system  consisting  of  two  single  spherical  refracting  surfaces, 
whose  focal  lengths,  in  terms  of  the  above  data,  will  be  expressed  as 
follows : 


1 
I 


—  J2  — 


n  —  i  n  —  i 

nrn  r.. 


n  —  i  n  —  i 


In  order  to  determine  the  positions  of  the  Focal  Points  F,  E'  and 
the  magnitudes  of  the  Focal  Lengths  /,  e'  of  the  Lens,  we  shall  employ 
formulae  (136),  in  which  the  determination-data  are  the  Focal  Lengths 
/„  e[  and/2,  e'2  of  the  two  partial  systems  and  the  "interval"  A  between 
them.  This  latter  magnitude  is  denned  as  follows: 

A  =  E(F2  =  E(A,  +  A,A2  +  A2F2, 
that  is, 

A  =  «;-/2  +  d;  (165) 

and,  accordingly,  expressed  in  terms  of  the  original  data,  this  magni- 
tude is  as  follows: 


=  n(rt-rj+d(n-i)  =       N 

n  —  i  (n  —  i) 

where 

N  =  (n  -  i) {n(r2  -  rj  +  d(n  -  i)}  (167) 

denotes  a  constant  of  the  Lens. 


§  198.]  Lenses  and  Lens-Systems.  275 

198.  (i)  The  abscissa  of  the  Focal  Points  F,  E'  of  the  Lens  with 
respect  to  Flt  E2,  respectively: 

These  are  obtained  immediately  from  the  first  two  of  formulae  (136) 
as  follows: 


(I68) 


(2)   The  Focal  Lengths  f,  e'  of  the  Lens: 

Likewise,  from  the  last  two  of  formulae  (136),  we  obtain: 


T  FA=f  =          =-e'  =  A'E';  (169) 

where  A,  A'  designate  the  positions  on  the  optical  axis  of  the  two 
Principal  Points  of  the  Lens.  We  see  that  in  every  Lens,  surrounded 
by  the  same  medium  on  both  sides,  the  Focal  Lengths  f,  e'  are  equal  in 
magnitude,  but  opposite  in  sign.  (This  is  true  of  an  optical  system 
consisting  of  any  number  of  spherical  surfaces,  provided  %  =  rim\ 
see  §  193.) 

Hence,  also,  the  Nodal  Points  N,  N'  of  the  Lens  coincide  with  the 
Principal  Points  A,  A',  respectively;  which  is  characteristic  likewise 
of  any  optical  system  for  which  the  media  of  the  incident  and  emergent 
rays  are  identical  (§  193). 

(3)  Abscisses  of  the  Focal  Points  F,  E'  of  the  Lens  referred  to  the 
vertices  Alt  A2,  respectively: 

Since 

A.F^A.F.+F.F^  -A+F.F, 

A2Ef  =  A2E'2  +E;E'  =  -  «;+£;#, 

we  derive,  by  means  of  formulae  (164)  and  (168),  the  following  form- 
ulae for  locating  the  positions  of  F  and  E'  : 

Nr,  +  n(n-  i}r\  Nr2  -  n(n  -  i)r\ 


(4)  Abscissce  of  the  Principal  Points  A,  A'  of  the  Lens,  reckoned 
from  the  vertices  Alt  A2,  respectively: 
Since 

A,A  =  A^F  +  FA,     A2Af  =  A2E'  +E'A', 

formulae  (169)  and  (170),  together  with  formula  (165),  give  the  fol- 
lowing formulae  for  determining  the  positions  of  A,  A': 

n  —  i  ,          n  —  i  A}A       r^  . 

A,A  =  -  —^-  r,d,     A2A=--^r-r2d,     -£-g  =  -  ;     (171) 


276  Geometrical  Optics,  Chapter  VIII.      .  [  §  199. 

whence  we  see  that  the  abscissae  of  the  Principal  Points  are  in  the  same 
ratio  to  each  other  as  the  radii  of  the  surfaces  of  the  Lens. 

The  distance  A  A'  between  the  two  Principal  Points  may  be  ex- 
pressed as  follows: 

AAf  =  AA,  +A,A2  +  A2A', 
whence  we  obtain: 


(I72) 


199.    Character  of  the  Different  Forms  of  Lenses. 

An  inspection  of  formulae  (167)  and  (169)  will  show  that  the  sign 
of  the  Focal  Length  /,  which  determines  the  character  of  the  Lens, 
depends  not  only  on  the  magnitudes  and  signs  of  the  radii  rlt  r2,  but 
also  on  the  thickness  d  of  the  Lens.  It  will  depend  also  on  whether  n 
is  greater  or  less  than  unity,  but  in  the  following  discussion  it  will  be 
assumed  that  the  Lens  is  of  the  type  of  a  glass  lens  in  air,  that  is, 

n  —  i  >  o. 

Evidently,  with  this  assumption,  the  magnitude  denoted  by  N  will  be 
greater  than,  equal  to,  or  less  than,  zero,  according  as 


Now  d  itself  is  always  positive  ;  and  hence  in  any  form  of  Lens,  for 
which  r^  —  r2  is  negative,  the  two  lower  signs  in  formula  (173)  cannot 
possibly  occur,  so  that  for  any  Lens  of  such  form,  the  sign  of  N  will 
necessarily  be  negative. 

How  the  sign  of  the  Focal  Length  /  of  the  Lens  depends  on  the 
magnitude  of  the  thickness  d,  will  be  apparent  in  the  following  clas- 
sification of  the  different  forms  of  Lenses. 

(i)  Biconvex  Lens  (r^  >  o,  r2  <  o).  In  a  Biconvex  Lens  the  radii 
rlt  r2  have  opposite  signs,  and  hence  the  Focal  Length 

f      nrfo 

J~     N 

of  a  Biconvex  Lens  is  positive,  so  long  as  the  thickness 

,  -  r2) 


d  < 


n  —  i 


so  that  a  Biconvex  Lens  whose  thickness  d  does  not  exceed  this  limiting 
value  is  a  convergent  Lens.     This  is  the  usual  character  of  a  Biconvex 


199.] 


Lenses  and  Lens-Systems. 


277 


glass  Lens  in  air,  an  example  of  which  is  shown  in  Fig.  101,  where 
ri  —  +  I0>  r2  =  —  15  and  d  =  +  3.  For  comparatively  small 
values  of  d,  as  here,  the  Principal  Points  A,  A'  are  situated  within  the 
Lens  itself.  If  the  Lens  is  made  thicker,  the  two  Principal  Points 


FIG.  101. 
CONVERGENT  BICONVEX  GLASS  LENS  IN  AIR. 

=  3/2;    n  =  AiCi  =  +  W',    ys  =  ^2C2  =  — 15;    d  =  AiA2=  +  3;    AiF=  — 11.66; 
+ 11.25;    AiA  =  +  0.833;    A2A'  =  — 1.25;   f=  FA  =  —  e»  =  A'E'  =  +  12.5. 

will  approach  nearer  to  each  other,  until  when  d  attains  the  value 
d  =  rl  —  r2,  so  that  the  two  surfaces  of  the  Lens  have  a  common 
centre,  the  Principal  Points  A,  A'  coincide  with  each  other  at  this 
common  centre.  Fig.  102  shows  a  Biconvex  Lens  with  concentric 
surfaces;  such  a  Lens,  made  of  glass  and  surrounded  by  air,  will  be 
convergent.  Here,  likewise,  belongs  the  Spherical  Lens,  character- 


E' 


FIG.  102. 

CONVERGENT  BICONVEX  GLASS  LENS  IN  AIR:  SPECIAL  CASE—  Two  SURFACES  CONCENTRIC. 
«=3/2;    n=+5;    rz  =  —  3;    d=  +  8;    A\F=  —  0.625;    AzE'  =  +  2.625  ; 

+  5;    AzAf  =  AzCa  =  A2Ci  =—  3;   f=FA  =  —e'  =  A'E'=  +5.625. 


ized  by  the  value  d  =  rl  —  r2  =  2^.  The  Spherical  Lens  is  also  to  be 
regarded  as  a  particular  case  of  the  Equi-Biconvex  Lens  (rl  =  —  r2, 
r1  >  o).  A  Spherical  Lens  is  shown  in  Fig.  103. 

If  we  suppose  the  thickness  of  the  Biconvex  Lens  to  be  greater 
than  (rl  —  r2),  the  Lens  continues  at  first  to  be  convergent,  but  the 
Primary  Principal  Point  A  will  lie  now  beyond  (or  to  the  right  of) 
the  Secondary  Principal  Point  A';  and  as  the  thickness  d  is  increased, 
these  points  separate  farther  and  farther  from  each  other,  so  that  at 
length  we  shall  find  the  Secondary  Principal  Point  A'  in  front  of  the 


278  Geometrical  Optics,  Chapter  VIII.  [  §  199. 

Lens  and  the  Primary  Principal  Point  A  beyond  the  Lens,  the  Lens 
still  being  convergent.     And  when  d  attains  the  value 

d  =  nfa  -  r2)/(n  -  i), 


FIG.  103. 
CONVERGENT  EQUIBICONVEX  GLASS  I<ENS  IN  AIR  :  SPECIAL  CASE  —  SPHERICAL  I<ENS. 


n  =  3/2;    n  =  AiCi  =  —  r*  =  CA2  =  +3  ;    d  =  AiAz  =  +  6  ;    AiF=E'A>i=  +  1.5; 
A\A  =••  AiC=  A'  A*  =  CAz  =  +  3  ;  /=  FA  =  —  e?  =  A'  Ef  —  +  4.5. 

the  Biconvex  Lens  becomes  a  Telescopic  Optical  System,  with  its 
Focal  Planes,'  and  its  Principal  Planes  also,  at  infinity.  The  case  of 
a  Telescopic  Biconvex  glass  Lens  in  air  is  shown  in  Fig.  104;  for 


FIG.  104. 

TELESCOPIC  BICONVEX  GLASS  I^ENS  IN  AIR. 
n-3/2;    n  =  ^lG  =  +3;    n  =  A-iGi  =  —  2;    rf  =  ^1^2  =  +  15;    AiO  =  +9;   /  =  —  ^  =  o>. 

which  the  determination-constants  have  the  following  values: 
PI  =  +  3,  r2=  — 2,  d  =  +15.  The  Optical  Centre  O  of  this  Lens 
coincides  with  the  Focal  Point  E{  of  the  first  surface  of  the  Lens  and 
with  the  Focal  Point  F2  of  the  second  surface  of  the  Lens. 

And,  finally,  in  case  d  >  n(rl  —  r2)/(n  —  i),  a  Biconvex  glass  Lens 
in  air  will  be  divergent  (/  <  o).  But  no  matter  how  great  the  thick- 
ness d  becomes,  we  shall  find  that  the  Focal  Point  F  of  a  Biconvex 
Lens  lies  always  in  front  of  the  Lens. 

(2)  Biconcave  Lens  (r2  >  o  >  rj.  Here  also,  as  in  the  case  of  the 
Biconvex  Lens,  the  radii  of  the  two  surfaces  have  opposite  signs,  rl 
being  negative  and  r2  being  positive.  Hence,  assuming  that  n  is 
greater  than  unity,  we  find  in  the  case  of  a  Biconcave  Lens  that  the 
constant  N  is  always  positive,  and,  therefore,  /  is  negative;  so  that 


199.[ 


Lenses  and  Lens-Systems. 


279 


a  Biconcave  glass  Lens  in  air  is  always  a  divergent  Lens.  The  Prin- 
cipal Points  A,  A'  of  a  Biconcave  Lens  lie  always  in  the  interior  of 
the  Lens,  the  Primary  Principal  Point  A  being  situated  in  front  of  the 
Secondary  Principal  Point  A'.  Fig.  105  shows  a  Biconcave  glass  Lens 


F 


FIG.  105. 
DIVERGENT  BICONCAVE  GLASS  I^ENS  IN  AIR. 

— 10;    y&  =  ^2C2=  +  15;    d  =  AiAn  =  +  3;    AiF 
+0.77;    A*Ar  =  — 1.154;  f=FA  =—  d  ^A'E1  =  — 11.54. 

in  air,  for  which  the  constants  have  the  following  values  :  rl  =  — •  10, 

r2  =  +  J5»  &  =  +  3- 

In  an  Equi-Biconcave  Lens  we  have  r2  =  —  rlf  r2  >  o. 

(3)   Lews  with  One  Surface  Plane.     In  this  case,  therefore,  one  of  the 
radii  rlt  r2  is  infinite. 

If  the  first  surface  is  the  plane  surface,  then  ^  =  00,  and  we  find: 


-  I ' 

so  that  the  character  of  the  Lens  depends  on  the  sign  of  the  curvature 
of  the  curved  surface.  Thus,  for  example,  in  a  Piano- Convex  Lens 
(rl  =  oo,  r2  <  o),  /  is  positive,  and  the  Lens  is  a  convergent  Lens. 
On  the  contrary,  in  a  Piano-  Concave  Lens  (rl  =  oo ,  r2  >  o) ,  /  is  negative, 
and  the  Lens  is  a  divergent  Lens.  (In  these  statements  it  is  assumed, 
as  always  in  this  discussion,  that  n  >  i). 
For  7^  =  00,  we  find  also: 


nr 


(n  — 


«*I*  /  \  *  •*  *•  -7  *  .V  * 

n(n  —i)  n  —  i 

A^A  =  d/n,     A2A'  =  o. 

When  one  surface  of  the  Lens  is  plane,  one  of  the  Principal  Points 
will  coincide  with  the  vertex  of  the  curved  surface. 

The  diagrams  (Figs.  106  and  107)  show  the  cases  of  a  Piano-Convex 
Lens  and  of  a  Piano-Concave  Lens  (n  =  3/2).  The  two  Lenses  are 


280 


Geometrical  Optics,  Chapter  VIII. 


[  §  199. 


represented  as  having  the  same  thicknesses,  and  the  absolute  value 
of  the  radius  of  the  curved  surface  is  the  same  for  both  Lenses.  In 
the  figures  the  first  surface  is  the  plane  surface  (rl  =  oo);  but  if  the 
light  is  supposed  to  go  from  right  to  left,  so  that  the  curved  surface 


E' 


FIG.  106. 
PLANO-CONVEX  GLASS  I^ENS  IN  AIR.    This  I^ens  is  always  Convergent. 


3/2; 


is  the  first  surface,  the  figures,  except  for  certain  obvious  changes  in 
the  letters,  will  be  correct. 

(4)  Concavo-  Convex,  or  Convexo-  Concave,  Lens.  In  a  Lens  of  this 
form  the  two  radii  rlt  r2  have  the  same  sign,  and,  hence,  the  sign  of  the 
Focal  Length  /  =  nr^/N  will  be  the  same  as  the  sign  of  the  constant 
N.  If,  therefore,  N  is  positive,  the  Lens  will  be  convergent;  if  AT" 
is  negative,  the  Lens  will  be  divergent;  and  if  N  =  o,  the  Lens  will 


E' 


FIG.  107. 

PLANO-CONCAVE  GLASS  I^ENS  IN  AIR.    This  I^ens  is  always  Divergent. 

n  =  3/2;    n**AiCi=  «;  rz  =  AzGi=  +6;    d=AiAt=+3;    AiF~+14;    AsE'^-12; 

AiA  =  +2;    A<tA'  =  Q;   f=  FA  =  —  ^  =  A' E'  =  — 12. 

be  telescopic.  According  to  (173),  the  sign  of  N  will  depend  on  the 
value  of  d. 

Let  us  suppose  that  both  radii  are  positive,  so  that  the  first  surface 
of  the  Lens  is  convex  (rl  >  o)  and  the  second  surface  is  concave 
(r2  >  o).  It  will  be  necessary  to  consider  only  this  case,  since  we 
have  merely  to  suppose  that  the  direction  of  the  light  is  reversed  in 
order  to  obtain  the  opposite  case. 

Accordingly,  assuming  that  both  radii  are  positive,  we  have  to 
consider  the  following  three  cases  of  the  Convexo-Concave  Lens: 

(a)  Positive  Meniscus  (r2  >  rx  >  o). 

Since  in  this  case  (rl  —  r2)  is  negative,  and  since  d  is  always  positive, 


§  199.]  Lenses  and  Lens-Systems.  281 

it  follows  that  d  >  n(rl  —  r2)/(n  —  i),  and,  therefore,  N  is  positive. 
Hence,  a  Lens  of  this  form,  called  a  "Positive  Meniscus",  is  always 
a  convergent  Lens  (Fig.  108).  It  will  be  remarked  that  the  Primary 


E' 


FIG.  108. 

POSITIVE  MENISCUS  (GLASS  LENS  IN  AIR).    This  Lens  is  always  Convergent. 
n  =  3/2;    n  =  AiCi=+6;    r2=  ,42C2=  +  12;    d  =  A\Az=  +2;    A\F=—  22.8;    AzE'  = 
A\A=—1.2;    A-iA'  =  —2A\   f=  FA  =—  e'  =  A' E'  =  +21.6. 

Principal  Point  A  lies  to  the  left  of  the  vertex  Alt  and  the  Secondary 
Principal  Point  A'  lies  to  the  left  of  the  vertex  A2\  and  that  the  line- 
segment  A  A'  is  always  positive. 

(b)  The  case  when  rx  >  r2  >  o.  In  this  case  (rx  —  r2)  is  positive, 
so  that  the  Lens  may  be  divergent,  convergent  or  telescopic,  depend- 
ing on  the  value  of  the  ratio  d/fa  —  r2). 

The  most  common  case  under  this  head  is  that  for  which 

d  <  n(rl  —  r2)/(n  -  i). 

When  this  is  the  case,  we  have  a  divergent  Lens,  called  a  "Negative 
Meniscus"  (Fig.  109).  The  Principal  Points  A,  A'  lie  beyond  (that 


FIG.  109. 

NEGATIVE  MENISCUS  (GLASS  LENS  IN  AIR). 
3/2;    n  =  ^iCi=+12;    ^  =  ^2C2=  +  6;    d=  A\Az=  +2;    AiF= 


is,  to  the  right  of)  the  vertices  Alt  A2,  respectively.     If  d  =  rl  —  r2, 
the  two  Principal  Points  coincide  at  a  point  which  is  also  the  com- 
mon centre  of  the  two  surfaces  of  the  Lens  (Fig.  no).     An  infinitely 
thin  Lens  of  this  kind  is  not  divergent,  but  telescopic. 
Again,  if  when  rx  >  r2  >  o,  we  have  also 

n(r,  -  r2) 
a  =  —     -  . 

n  -  i 


282 


Geometrical  Optics,  Chapter  VIII. 


[  §  199. 


the  Lens  will  be  of  the  kind  represented  in  Fig.  1 1 1 ,  where  the  constants 
have  the  following  values:  n  —  3/2,  rl  =  -f  12,  r2  —  +  6,  d  =  +  18; 


FIG.  110. 

CONVEXO-CONCAVE  GLASS  L.ENS  IN  AIR  :  SPECIAL  CASE  OF  NEGATIVE  MENISCUS.  Two  surfaces 
of  I<ens  have  the  same  centre:  Lens  is  Divergent.  (Focal  Point  .F  not  shown  in  the  diagram;  it 
lies  far  to  the  right.) 

n  =  3/2;     n  =  AiCi  =  Aid  =  A\A  =  AiA'  =  +5  ;     ^  =  ASC2  =  AiC\  =  AzA  =  AZA'  =  +  3  ; 
n  —  rz=  +2;    AiF=+27.5;    AZE'  =  —19.5  ;/=  FA  =  —  a*  =  A'  E'  =  —22.5. 


whence  we  find  /  =  —  e'  —  <».     This  type  of  Lens  may,  therefore, 
be  called  a  "Telescopic  Meniscus". 

As  d  increases  from  the  value  d  =  r^  —  r.2  to  the  value 

d  =  n(rl  -  r2)/(n  -  i), 

the  Principal  Points,  which,  as  we  saw  above,  were  coincident,  sepa- 
rate farther  and  farther  from  each  other,  both  moving  along  the  optical 


FIG.  111. 

CONVEXO-CONCAVE  GLASS  I^ENS  IN  AIR:  SPECIAL  CASE—  TELESCOPIC  MENISCUS. 
«  =  3/2;    n  =  ^iCi=CiC2=+12;    ra  =  A*Cz=  CiAz=  +  6;   /=—/=». 


axis  in  the  positive  direction  of  that  axis,  but  A'  keeping  ahead  of  A 
until  they  both  arrive  together  at  infinity. 

And,  finally,  if,  when  ^  >  r2  >  o,  we  have  also 

d  >  n(r^  -  r2)/(n  -  i), 
as  in  the  case  of  the  Lens  represented  in  Fig.  112,  where  the  constants 


FIG. 112. 
CONVERGENT  MENISCUS  (n  >  n  >  0) :  GLASS  LENS  IN  AIR. 

;    n  =  AiCi=>  +3;  n  =  AzCz=  +2;    d  =  A\A*=  +6;    A\F=—  24; 
=+4;    AiA  =  -\2;    AsAf  =  —8;   f=FA=—S  =  A'E'=+l2. 


200.] 


Lenses  and  Lens-Systems. 


283 


have  the  following  values:   n  —  3/2,  rL  =  +3,  r2  =  -f  2,  d  =  +  6, 

the    Lens  will   again    be   a    convergent 

Lens,  and  now  the   Principal  Points  A, 

A'  will  lie  in  front  of  the  vertices  Alt 

A 2,  respectively,  and  A  will  lie  in  front 

otA'. 

(c)  The  last  case  to  be  considered  is 
the  case  when  rl  —  rz>  o.  In  this  form 
of  Lens,  sometimes  called  "Lens  of  Zero- 
Curvature",  the  curvatures  of  the  two 
surfaces  are  equal;  and  since  rlf  r2  have 
the  same  sign,  and  N  is  positive,  the 
Focal  Length  /  is  positive,  so  that  this 
Lens  is  always  convergent.  The  diagram 
(Fig.  113)  represents  the  case  of  a  Lens 


FIG.  113. 

CONVERGENT  CONVEXO-CONCAVE 
GLASS  I,ENS  IN  AIR:  Special  Case  — 
Two  surfaces  have  equal  curvature : 
Called  "  I^ens  of  Zero  Curvature.'' 


of  Zero-Curvature,  determined    by  the  values: 
=  6rl/d.     Obviously,  in  this  Lens  we  have 


VI 


n 


3/2, /=  ~e' 


In  the  limiting  case  when  d  =  o,  this  Lens  will  be  an  infinitely  thin 
telescopic  Lens. 

ART.  57.     THIN  LENSES. 

200.  Practically  speaking,  the  thickness  of  a  Lens  is  almost  always 
small  in  comparison  with  the  other  linear  constants  of  the  Lens.  And 
(except  in  the  case  of  the  so-called  "Lens  of  Zero-Curvature",  for 
which  rl  =  r2)  the  term  (n  —  i)d  which  occurs  in  the  expression  of  the 
constant  N  is  generally  quite  small  in  comparison  with  the  other 
term  n(r2  —  rj.  The  value  of  N  may  be  written: 


and,  hence,  if  we  neglect  terms  involving  powers  of  d  higher  than  the  first, 
we  have: 

JL  =  i  f    _  (n  -  i)d  \ 

N      n(n  -  i)(r2  -  rj  [  *       n(r2  -  rj  }' 

Substituting  this  value  of  i/N  in  formulae  (169)  and  (171),  we  obtain 
the  following  approximate  formula  of  Thin  Lenses: 


284  Geometrical  Optics,  Chapter  VIII.  [  §  202. 

(n  -  i)d 

[' 

(174) 


=  —  e'  =  7—   — r^ r  1 1  - 


n(r2  -  r, 

rA 

A9A'  =  -  • 


-*>' 

These  formulae  are  useful  for  approximate  determinations  of  the  magni- 
tudes of  the  Focal  Lengths  and  of  the  positions  of  the  Principal  Points. 
201.  Infinitely  Thin  Lenses.  If  the  Lens  is  infinitely  thin,  the 
formulae  above  may  be  still  further  simplified  by  putting  d  =  o.  Thus, 
we  obtain: 

,  rir2 

d  =  AiAt  =  o,    /  =  —  e'  —  , — _    .  , 

(175) 


These  formulae,  which  are  identical  with  those  formerly  obtained  in 
Chapter  VI,  need  no  further  remark  here. 

ART.  58.     LENS-SYSTEMS. 

202.  Consider  a  compound  system  consisting  of  two  Lenses  with 
their  optical  axes  in  the  same  straight  line,  and  let  Alt  A[  and  Flt  E{ 
designate  the  positions  of  the  Principal  Points  and  of  the  Focal  Points, 
respectively,  of  the  first  Lens;  similarly,  let  A2,  A'2  and  F2,  E2  designate 
the  positions  of  the  Principal  Points  and  of  the  Focal  Points,  respect- 
ively, of  the  second  Lens.  Thus, 

Wi  =/i  =  -  e'i  =  A{E{,     F2A2  =/a  =  -e'2  =  A2E2] 

where  flt  e(  and  /2,  e'2  denote  the  Focal  Lengths  of  the  two  Lenses. 
Moreover,  here  let  us  put 

A{A2  =  d,     E(F2  =  A. 
Then,  since 

A  =  E\A(  -f  A(A2  +  A2F2, 
we  have: 

A=  -(/,+/,-<*). 
•s 

Finally,  let  F,  E'  and  A,  A'  designate  the  positions  of  the  Focal  Points 
and  of  the  Principal  Points,  respectively,  of  the  compound  system  of 
the  two  Lenses,  and  let/,  e'  denote  the  Focal  Lengths  of  the  compound 
system.  Then,  by  processes  entirely  similar  to  those  employed  above 
in  Art.  56,  we  derive  the  following  system  of  formulae  for  an  Optical 


§  204.]  Lenses  and  Lens-Systems.  285 

System  composed  of  two  Lenses: 


\+f*-d' 

/?  E'E'  =  -        f 


-d'  /,+/,-<*' 

'-^-      A'E'- 

,     ,  ,/io-/-*  


(176) 


1  J* 

Jl 

Thus,  being  given  the  two  Lenses  and  their  positions  relative  to  each 
other,  we  can,  by  means  of  the  above  formulae,  determine  completely 
the  compound  system. 

203.  If,  instead  of  two  Lenses,  we  had  Two  Systems  of  Lenses,  the 
formulae  (176)  can  be  employed  to  determine  the  compound  system, 
provided  the  letters  with  the  subscript  i  and  the  letters  with  the  sub- 
script 2  be  understood  as  applying  to  the  first  and  second  systems  of 
Lenses,  respectively. 

204.  A  case  of  considerable  interest  is  an  Optical  System  composed 
of  Two  Infinitely  Thin  Lenses.     Since  the  two  Principal  Points  of  an 
infinitely  thin  Lens  coincide  at  the  optical  centre  of  the  Lens,  the 
letters  Al  and  A'2J  as  employed  in  formulae  (176),  will  designate  for 
this  case  the  positions  of  the  optical  centres  of  the  two  Lenses,  and, 
therefore, 

denotes  now  the  distance  of  the  second  Lens  from  the  first.  If  the 
two  infinitely  thin  Lenses  are  in  contact  (d  =  o) ,  we  find : 

!//=    I//1+I//,, 

in  agreement  with  the  general  formula  (106). 

Assuming,  for  the  sake  of  simplicity,  that  the  optical  system  con- 
sists of  two  infinitely  thin  Lenses,  we  may  discuss  formulae  (176) 
briefly,  as  follows: 

(a)  Suppose  that  both  Lenses  are  convergent  (/t  >  o,  /2  >  o.)  If  the 
two  Lenses  are  in  contact  (d  =  o) ,  we  have : 


and,  consequently,  /  >  o.     But  this  is  the  smallest  positive  value  of/; 


286  Geometrical  Optics,  Chapter  VIII.  [  §  204. 

so  that  as  we  increase  the  distance  d  between  the  two  Lenses,  the 
resulting  system  is  less  and  less  convergent;  until,  when  d  =  /x  -f/2, 
we  have  /  =  oo,  in  which  case  the  compound  system  is  telescopic. 
If  we  continue  to  separate  the  Lenses  still  farther,  we  have  at  first  a 
feebly  divergent  system;  but  the  divergence  increases  as  d  is  made 
greater  and  greater. 

(b)  In  case  both  Lenses  are  divergent  (/i  <  o,/2  <  o),  we  have  always 
/  <  o,  so  that  the  compound  system  will  be  divergent.     The  diver- 
gence will  be  greatest  when  the  two  Lenses  are  in  contact,  and  will 
decrease  as  the  Lenses  are  separated  farther  and  farther. 

(c)  Finally,  suppose  that  one  of  the  Lenses  is  convergent,  and  the 
other  divergent.     For  example,  let  us  assume  that  /L  >  o  and  /2  <  o. 
In  this  case  the  compound  system  will  be  divergent,  if  d  <  (/t  -f  /2) ; 
convergent,  if  d  >  (/t  +  /2) ;  and  telescopic,  if  d  =  fa  +  /2.     Since/!,  /2 
have  opposite  signs,  there  are  two  cases  here  to  be  considered,  as 
follows : 

ist,  The  case  when  (fa  +  /2)  <  o:  that  is,  the  absolute  value  of  the 
Focal  Length  of  the  convergent  Lens  is  less  than  that  of  the  divergent 
Lens.  Since  d  is  essentially  positive,  the  only  possibility  here  is 
d  >  (fa  +  /2),  and  hence  this  system  will  also  be  convergent.  The 
greatest  value  of  /  is  obtained  by  placing  the  two  Lenses  in  contact 
(d  =  o);  and  as  the  Lenses  are  separated  farther  and  farther  apart, 
the  convergence  increases. 

2nd,  The  case  when  (fa  +  /2)  >  o:  that  is,  the  absolute  value  of  fl 
is  greater  than  that  of /2.  When  the  two  Lenses  are  in  contact  (d  =  o), 
the  system  is  divergent  and  /  has  its  least  negative  value.  As  d  in- 
creases, the  absolute  value  of  /  increases,  its  sign  remaining  negative ; 
until,  when  d  =  /i  +  /2,  /  is  infinite,  and  the  system  is  a  telescopic 
system.  For  values  of  d  greater  than  (fl  +  /2),  the  sign  of/  will  be 
positive,  and  the  system  will  be  convergent,  the  convergence  increasing 
with  continued  increase  of  d. 


CHAPTER   IX. 

EXACT   METHODS  OF   TRACING  THE  PATH   OF  A  RAY  REFRACTED  AT 
A    SPHERICAL    SURFACE. 

ART.  59.     INTRODUCTION. 

205.  In  the  preceding  chapter  we  have  seen  how  an  ideal  image  is 
produced  by  a  centered  system  of  spherical  surfaces  so  long  as  the 
rays  concerned  are  the  so-called  "Paraxial  Rays"  which  are  all  con- 
tained within  the  infinitely  narrow  cylindrical  region  immediately 
surrounding  the  optical  axis  of  the  system.  In  this  case  to  a  homo- 
centric  bundle  of  incident  rays  corresponds  a  homocentric  bundle  of 
emergent  rays. 

But,  according  to  the  Wave-Theory  of  Light,  in  order  to  have  an 
optical  imagery,  a  mere  homocentric  convergence  of  the  rays  is  not 
sufficient.  This  theory  requires  not  only  that  the  wave-front  after 
the  light  has  traversed  the  optical  system  shall  be  spherical,  so  that 
the  rays  of  light  proceeding  originally  from  a  point  shall  meet  again 
in  a  point,  but  that  the  effective  portion  of  the  wave-surface  shall  be 
as  great  as  possible  in  comparison  with  its  radius,  which  means  that 
the  effective  rays  shall  constitute  a  wide-angle  bundle  of  rays  (see  §  45). 
Only  when  this  last  condition  is  complied  with  will  the  resultant  effect 
of  the  spherical  wave  be  reduced  approximately  to  a  point  at  the  centre, 
so  that  there  will  be  point-to-point  correspondence  between  object 
and  image. 

Moreover,  there  is  also  still  another  practical  reason  why  we  find 
it  necessary  to  use  wide-angle  bundles  of  rays  in  the  production  of 
an  image.  For  if  the  wide-angle  bundle  of  rays  is  a  condition  of  a 
distinct,  clear-cut  image,  it  is  equally  essential  for  the  production  of 
a  bright  image,  since  the  light-intensity  will  evidently  be  greater  in 
proportion  as  the  effective  portion  of  the  wave-surface  is  larger. 

Both  theoretically  and  practically,  therefore,  we  require  to  have  an 
optical  system  which  will,  if  possible,  converge  to  a  point  a  wide-angle 
homocentric  bundle  of  incident  rays,  so  that  not  merely  those  rays 
which  we  call  Paraxial  Rays  but  those  rays  which  have  finite  inclina- 
tions to  the  optical  axis  will  be  converged  again  to  one  and  the  same 
image-point.  Generally  speaking,  this  requirement  is  found  to  be 
impossible  of  fulfilment.  Indeed,  there  may  be  said  to  be  only  one 
actual  optical  system  which  perfectly  satisfies  the  condition  of  collinear 

287 


288  Geometrical  Optics,  Chapter  IX.  [  §  206. 

correspondence,  viz.,  the  Plane  Mirror;  which,  inasmuch  as  it  pro- 
duces only  a  virtual  image  without  magnification,  hardly  deserves  to 
be  ranked  as  an  "optical  instrument"  at  all.  The  "Pin-Hole  Camera" 
is  no  exception  to  this  statement,  because  only  when  the  aperture 
through  which  the  rays  enter  the  apparatus  is  a  mathematical  point 
will  there  be  strict  point-to-point  correspondence  of  object  and  image 
— even  then  assuming  that  there  were  no  exceptions  to  the  Law  of  the 
Rectilinear  Propagation  of  Light  such  as  we  encounter  in  Physical 
Optics. 

Instead  of  the  ideal  case  of  collinear  correspondence  of  Object- 
Space  and  Image-Space,  the  theory  of  optical  instruments  is  compli- 
cated by  numerous  practical  and,  for  the  most  part,  irreconcilable 
difficulties,  due  chiefly  to  the  so-called  "aberrations" — some  of  which 
are  aberrations  of  sphericity,  while  others  are  chromatic  aberrations — 
and  due  also,  in  a  less  degree,  to  the  assumptions  at  the  foundation  of 
Geometrical  Optics,  which,  as  we  have  pointed  out,  are  not  entirely 
in  accordance  with  the  facts  of  Physical  Optics.  It  is  not  our  purpose, 
however,  to  enter  into  a  discussion  of  these  questions  here,  as  they 
will  be  extensively  treated  in  subsequent  chapters  of  this  treatise. 
In  this  chapter  we  propose  to  investigate  the  path  of  a  ray  which 
makes  a  finite  angle  with  the  axis. 

ART.  60.     GEOMETRICAL  METHOD  OF  INVESTIGATING  THE  PATH  OF  A  RAY 
REFRACTED  AT  A  SPHERICAL  SURFACE. 

206.     Construction  of  the  Refracted  Ray. 

In  §  29,  we  showed  how  to  construct  the  path  of  a  ray  refracted  at 
a  surface  of  any  form,  and  that  method  is,  of  course,  applicable  to 
the  refraction  of  a  ray  at  a  spherical  surface.  The  following  elegant 
and  useful  construction  of  the  path  of  a  ray  refracted  at  a  spherical 
surface  was  first  given  by  THOMAS  YOUNG  in  his  lectures  on  Natural 
Philosophy.1  WEiERSTRAss,2  in  1858,  and  LiPPiCH,3  in  1877,  gave 
the  same  construction,  each  entirely  independently. 

Let  C  (Figs.  114  and  115)  designate  the  position  of  the  centre,  and 
let  r  denote  the  radius,  of  the  spherical  refracting  surface  n,  and  let 

1  A  course  of  lectures  on   Natural  Philosophy  and  the  Mechanical  Arts,  by  THOMAS 
YOUNG,  M.D.,  London,  1807  (two  volumes);  II,  p.  73,  Art.  425. 

2  See  article  by  K.  SCHELLBACH  entitled  "  Der  Gang  der  Lichtstrahlen  in  einer  Glas- 
kugel":   Zft.  phys.  chem.    Unt.,  1889,  II,  135. 

3F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme  an 
Kugelflaechen:  Denkschriften  der  kaiserl.  Akad.  der  Wissenschaften  zu  Wien,  xxxvn 
(1878),  pp.  163-192.  See  also  a  paper  published  by  F.  KESSLER  in  Wied.  Ann.  Phys., 
xv  (1882). 


§  206.] 


Path  of  Ray  Refracted  at  Spherical  Surface. 


289 


QB  represent  the  path  of  the  incident  ray  meeting  ^  at  the  point  B\ 
and  let  n,  n'  denote  the  absolute  indices  of  refraction  of  the  first  and 
second  medium,  respectively.  Concentric  with  the  spherical  refracting 
surface  M,  and  with  radii  equal  to  n'r/n  and  nr/n',  describe  two  spheri- 
cal surfaces  r,  T',  respectively.  Let  Z  designate  (as  shown  in  the 
diagram)  the  point  where  QB,  produced  if  necessary,  meets  the  auxil- 
iary spherical  surface  r.  Join  Z  by  a  straight  line  with  the  centre  C, 
and  let  Z'  designate  the  point  where  this  straight  line  intersects  the 


FIG. 114. 

YOUNG'S  CONSTRUCTION  OF  THE  PATH  OF  A 
RAY  REFRACTED  AT  A  SPHERICAL  SURFACE. 
The  figure  shows  the  case  when  the  surface  is 
convex  and  the  second  medium,  more  highly 
refracting  than  the  first  (nf  >  n).  If  the 
letters  Q  and  R' ,  Z  and  Z1  and  r  and  rf  are 
interchanged,  and  if  the  arrow-heads  are  re- 
versed, the  same  diagram  will  show  YOUNG'S 
Construction  for  the  case  when  the  ray  is  re- 
fracted at  a  concave  spherical  surface  into  an 
optically  less  dense  medium. 


FIG. 115. 

YOUNG'S  CONSTRUCTION  OF  THE  PATH  OF  A 
RAY  REFRACTED  AT  A  SPHERICAL  SURFACE. 
The  figure  shows  the  case  when  the  surface  is 
concave,  and  the  second  medium  more  highly 
refracting  than  the  first  (n'  >  n).  If  the 
letters  Q  and  R! ',  Z  and  Z*  and  t  and  T'  are 
interchanged,  and  if  the  arrow-heads  are  re- 
versed, the  same  diagram  will  show  YOUNG'S 
Construction  for  the  case  when  the  ray  is  re- 
fracted at  a  convex  surface  into  an  optically 
less  dense  medium. 


other  auxiliary  spherical  surface  rf .  The  path  BR'  of  the  refracted 
ray  is  determined  by  the  straight  line  which  joins  B  and  Z' '.  In 
making  this  construction,  we  must  be  careful  to  select  for  the  point 
Z  that  one  of  the  two  possible  points  of  intersection  of  the  incident 
ray  QB  with  the  spherical  surface  ju  which  will  make  the  piece  of  the 
incident  ray  which  lies  in  the  first  medium  and  the  piece  of  the  re- 
fracted ray  which  lies  in  the  second  medium  fall  on  opposite  sides  of 
the  incidence-normal  CB,  in  accordance  with  the  Law  of  Refraction. 
The  proof  of  the  construction  is  very  simple.  Since 

CZ  :  CB  =  CB  :  CZ'  =  n'  :n, 


20 


290  Geometrical  Optics,  Chapter  IX.  [  §  207. 

the  triangles  CBZ,  CBZ'  are  similar,  and,  hence,  Z  CBZ  =  Z  BZ'C. 
But  in  the  triangle  CBZ, 

sin  Z.CBZ      CZ       nf 

sin  Z.BZC~  CB  ~  nf 

and,  since  by  the  Law  of  Refraction 

sin  a/sin  a'  =  n'/n, 

where  a  =  Z  CJ5Z,  it  follows  that  Z.BZC  =  Z  C£Z'  =  «'  and, 
therefore,  BR'  is  the  path  of  the  refracted  ray. 

In  both  diagrams  (Figs.  114  and  115)  the  case  represented  is  that 
in  which  the  first  medium  is  less  dense  than  the  second  (n'  >  ri) ; 
but  by  a  suitable  change  of  the  letters  and  a  reversal  of  the  arrow- 
heads, the  same  diagrams  will  suffice  to  exhibit  the  case  when  the  ray 
is  refracted  into  the  less  dense  medium  (nr  <  ri).  In  this  latter  case 
the  spherical  surface  r  will  be  the  inner,  and  the  spherical  surface  r' 
will  be  the  outer,  of  the  two  auxiliary  spherical  surfaces;  thus,  in  this 
case,  a  ray  may  be  incident  on  the  spherical  refracting  surface  p  with- 
out meeting  at  all  the  auxiliary  surface  r;  which  means  that  such  a 
ray  will  be  totally  reflected. 

207.     "Aplanatic"  Pair  of  Points  of  a  Spherical  Refracting  Surface. 

The  first  point  to  be  remarked  in  connection  with  YOUNG'S  Con- 
struction is  the  extraordinary  property  of  every  pair  of  such  points 
as  Z,  Z'.  Any  straight  line  drawn  through  the  centre  C  of  the  spher- 
ical refracting  surface  will  determine  by  its  intersections  with  the 
auxiliary  spherical  surfaces  r,  rf  a  pair  of  points  Z,  Z',  at  distances 
from  C  equal  to  n'r/n,  nrfn',  respectively,  characterized  by  the  prop- 
erty that  to  a  homocentric  bundle  of  incident  rays  Z  corresponds  a 
homocentric  bundle  of  refracted  rays  Z'.  Moreover,  this  property 
is  entirely  independent  of  the  angular  opening  of  the  bundle  of  inci- 
dent rays,  and  is  true,  therefore,  of  a  bundle  of  rays  of  finite  aperture. 

The  pair  of  conjugate  points  Z,  Z',  which  lie  on  the  axis  of  the 
spherical  refracting  surface  (Fig.  1 1 6) ,  and  which  are  situated  as  above 
described,  are  called  the  "aplanatic"  pair  of  points  of  the  spherical 
refracting  surface ;  with  respect  to  these  points  the  spherical  refracting 
surface  is  an  "aberrationless"  surface. 

Since  rays  which  are  directed  towards  the  centre  C  enter  the  second 
medium  without  being  changed  in  their  directions,  the  point  Cmay  also 
be  regarded  as  a  pair  of  coincident  conjugate  points  (§44)  which  possess 
a  property  similar  therefore  to  that  of  the  aplanatic  points.  Moreover, 
each  point  on  the  surface  of  the  refracting  sphere  is  a  "self-correspond- 


§  207.]  Path  of  Ray  Refracted  at  Spherical  Surface.  291 

ing",  or  ''double",  point.     But  the  only  pair  of  such  points  that  are 
separated  is  the  pair  Z,  Z'. 

Since  Z  BZ'C  =  a,  Z  BZC  =  a',  it  follows  that  the  angles  of  in- 
clination to  the  axis  of  the  incident  and  refracted  rays  are  equal  to 
the  angles  of  refraction  and  incidence,  respectively;  so  that  the  aplan- 


FlG. 116. 
SO-CALLED  APLANATIC  (OR  ABERRATIONLESS)  POINTS  OF  A  REFRACTING  SPHERE. 

atic  points  are  likewise  characterized  by  the  fact  that  the  sines  of  the 
angles  of  inclination  to  the  axis  of  any  pair  of  conjugate  rays  crossing 
the  axis  at  Z  and  at  Z'  have  a  constant  ratio.  Another  way  pf  re- 
marking this  characteristic  property  of  the  aplanatic  pair  of  points 
of  a  spherical  refracting  surface  is  by  the  relation: 

BZJBZ'  =  n'/n. 
Moreover,  since 

CZ-  CZ'  =  r\ 

the  geometer  will  recognize  that  Z,  Z'  are  the  so-called  "inverse" 
points  with  respect  to  the  spherical  surface  of  radius  r,  which  are 
harmonically  separated  by  the  end-points  of  the  diameter  on  which 
they  lie. 

The  points  Z,  Z'  lie  always  on  the  same  side  of  the  centre  C  of 
the  spherical  refracting  surface,  so  that  whereas  the  rays  will  pass 
"really"  through  one  of  these  points,  the  corresponding  rays  will  pass 
"virtually"  through  the  other  point.  Thus,  one  of  the  spherical  sur- 
faces r,  rf  is  the  virtual  image  of  the  other. 


292  Geometrical  Optics,  Chapter  IX.  [  §  208. 

The  circle  of  contact,  in  which  the  tangent-cone,  drawn  from  that 
one  of  the  points  Z,  Z'  which  lies  outside  the  spherical  refracting 
surface,  touches  this  surface,  divides  it  into  two  portions,  and  the 
incidence-point  B  lies  always  on  the  greater  of  these  two  portions  of 
the  surface. 

In  the  case  of  Reflexion  at  a  Spherical  Mirror  (n'  —  —  n),  YOUNG'S 
Construction  evidently  fails.  A  Spherical  Mirror  has  no  pair  of  "apla- 
natic"  points  corresponding  to  Z,  Z';  or,  more  correctly  speaking,  the 
points  Z,  Z'  coincide  at  the  vertex  of  the  mirror. 

208.     Spherical  Aberration. 

In  general,  however,  a  homocentric  bundle  of  rays  incident  on  a 
spherical  refracting  surface  will  not  be  homocentric  after  refraction. 
Consider,  for  example,  a  bundle  of  rays  diverging  from  a  point  L 
(Fig.  117),  and  incident  directly  on  a  Spherical  Refracting  Surface, 
so  that  the  chief  ray  of  the  bundle  is  directed,  therefore,  towards  the 
centre  C.  Since  there  is  symmetry  around  LC  as  axis,  it  will  be 
sufficient  to  trace  the  paths  of  those  rays  which  lie  in  a  meridian  section 
of  the  bundle,  for  example,  the  section  made  by  the  plane  of  the  dia- 


FIG.  117. 

SPHERICAL  ABERRATION.  Whereas  the  incident  rays  all  cross  the  axis  of  the  spherical  surface 
at  one  point  L,  the  corresponding  refracted  rays  cross  the  axis,  in  general,  at  different  points  L' 
L",  etc. 

gram ;  for  it  is  obvious  that  the  entire  bundle  of  rays  may  be  regarded 
as  generated  by  the  rotation  of  this  meridian  pencil  around  the  chief 
ray  L  C  as  axis. 

If  L  B  is  an  incident  ray,  and  BL'  the  corresponding  refracted  ray 
meeting  the  straight  line  LC'm  L',  and  if  LBLf  is  revolved  around  L  C 
as  axis,  to  the  incident  rays  lying  on  the  surface  of  the  right-circular 
cone  CLB  will  correspond  a  system  of  refracted  rays  lying  on  the 
surface  of  the  right-circular  cone  CL'B. 

The  position  of  the  point  L'  can  be  seen  to  depend,  in  general,  on 
the  slope  of  the  incident  ray  LB,  so  that  different  rays  of  the  pencil 


208.] 


Path  of  Ray  Refracted  at  Spherical  Surface. 


293 


of  incident  rays  L  will  determine  different  positions  of  the  point  L' '. 
Accordingly,  whereas  all  the  rays  of  the  bundle  of  incident  rays  L 
will  be  grouped  in  cones,  which  have  a  common  vertex  at  L,  the 


E' 


FIG.  118. 

SPHERICAL  ABERRATION.     Case  when  a  Pencil  of  Parallel  Incident  Rays  is  Refracted  at  a 
Spherical  Surface. 

corresponding  refracted  rays  will  be  grouped  in  cones,  which,  while  they 
have  all  a  common  axis  L  C,  will,  in  general,  have  different  vertices  L' '. 
This  variation  of  the  position  of  the  point  L'  corresponding  to  a  fixed 


FIG. 119. 

SPHERICAL  ABERRATION.    Case  when  a  Pencil  of  Parallel  Incident  Rays  is  Refracted  through 
an  Kqui-biconvex  I,ens  (glass  lens  in  air). 

position  of  the  point  L  is  called  Spherical  Aberration  (see  §  260) :  which 
will  be  treated  at  length  in  a  special  chapter  devoted  to  that  subject. 


294 


Geometrical  Optics,  Chapter  IX. 


[  §  209. 


The  diagram  (Fig.  118)  shows  the  case  of  a  meridian  pencil  of  in- 
cident rays  parallel  to  the  axis  of  the  spherical  refracting  surface. 
The  paths  of  the  refracted  rays  have  been  traced  by  YOUNG'S  Con- 
struction. The  outermost  ray  of  the  pencil  is  refracted  so  as  to  cross 
the  axis  at  a  point  marked  Z/,  whereas  a  Paraxial  Ray  will  be  refracted 
to  the  Focal  Point  Ef  of  the  Image-Space.  The  line-segment  E'  U  is  a 
measure  of  the  so-called  Longitudinal  Aberration  along  the  axis. 

Fig.  119  shows  in  the  same  way  the  Longitudinal  Aberration  along 
the  axis  of  an  Equi-Biconvex  Glass  Lens  in  Air. 


TRIGONOMETRIC  COMPUTATION    OF   THE    PATH   OF  A  RAY  OF  FINITE 

INCLINATION  TO  THE  AXIS,  REFRACTED  AT  A 

SINGLE  SPHERICAL  SURFACE. 


CASE  I. 


WHEN  THE  PATH  OF  THE  RAY  LIES  IN  A  PRINCIPAL  SECTION  OF  THE 
SPHERICAL  REFRACTING  SURFACE. 


ART.  61.      THE  RAY-PARAMETERS,  AND  THE  RELATIONS  BETWEEN  THEM. 

209.  Any  section  made  by  a  plane  containing  the  optical  axis  will 
be  called  a  Principal  Section  of  the  spherical  refracting  surface,  and 
under  Case  I  we  shall  consider  only  such  rays  as  lie  in  the  plane  of  a 
principal  section. 

In  the  diagram  (Fig.  120)  the  plane  of  the  paper  represents  a  prin- 
cipal section  of  the  spherical  refracting  surface,  the  centre  of  which 


FIG.  120. 
TRIGONOMETRIC  CALCULATION  OF  THE  PATH  OF  A  RAY  REFRACTED  AT  A  SPHERICAL  SURFACE  : 

CASE  WHEN  THE  RAY-PATH   LIES   IN  THE  PLANE  OF  A  PRINCIPAL  SECTION.      The  Straight  line  QB 

shows  the  path  of  the  incident  ray,  and  the  straight  line  BR!  shows  the  path  of  the  corresponding 
refracted  ray. 


AC=r,    AL  = 
£.  BCA  =  <*>, 


CU 


ALB=e,     £AL'B 


=  cf, 
BL 


BL' 


L  CBL  = 
I',    CH 


CBL'  =  a'. 


is  at  the  point  designated  by  the  letter  C.  Through  C  draw  a  straight 
line  in  the  plane  of  the  paper  meeting  the  spherical  surface  in  the 
point  A.  This  straight  line  we  shall  take  as  the  optical  axis,  so  that 
the  point  designated  by  A  will  be  the  vertex  of  the  surface.  Let  the 
straight  line  QB,  intersecting  the  optical  axis  at  a  point  L,  represent 


§  209.]  Path  of  Ray  Refracted  at  Spherical  Surface.  295 

the  path  of  a  ray  of  light  incident  on  the  spherical  refracting  surface 
at  the  point  B,  and  draw  the  radius  BC. 

We  shall  employ  here  pretty  nearly  the  same  letters  and  symbols 
as  were  used  in  Chapter  V,  with  such  changes,  however,  as  will  be 
necessary  in  order  to  distinguish  the  present  case  from  that  of  a  Parax- 
ial  Ray.  Moreover,  as  we  shall  also  have  to  introduce  symbols  for 
several  new  magnitudes,  and  as  the  relations  derived  below  will  be 
frequently  referred  to  in  the  course  of  this  work,  it  will  be  well  to 
define  clearly  the  precise  meaning  that  is  to  be  attached  to  each  of  the 
symbols  employed ;  which  we  therefore  proceed  to  do. 

i.  Notation  of  the  Linear  Magnitudes. 

The  abscissa  of  the  centre  C,  with  respect  to  the  vertex  A,  will  be 
denoted  by  r;  thus,  A  C  =  r. 

The  abscissae,  with  respect  to  the  vertex  A ,  of  the  points  designated 
by  L,  Z/,  where  the  ray  crosses  the  axis,  really  or  virtually,  before  and 
after  refraction,  will  be  denoted  by  v,  i/,  respectively ;  thus,  A  L  =  v, 
AL'  =  vf. 

The  abscissae,  with  respect  to  the  centre,  of  the  points  L,  L'  will 
be  denoted  by  c,  cf,  respectively;  thus,  CL  =  c,  CL'  —  c'\  and, 
consequently, 

c  =  v  —  r,     c'  =  v'  —  r. 

In  regard  to  the  signs  of  these  abscissae,  they  are  to  be  reckoned 
positive  or  negative  according  as  they  are  measured  in  the  positive  or 
negative  direction  of  the  axis:  the  positive  direction  of  the  axis  being 
determined  by  the  direction  of  the  incident  axial  ray  (§193). 

The  "ray-lengths"  are  the  segments  of  the  incident  and  refracted 
rays,  measured  from  the  point  of  incidence  B  to  the  points  L,  L'  where 
the  incident  and  refracted  rays,  respectively,  cross  the  axis.  These 
magnitudes  will  be  denoted  by  the  symbols  /,  /';  thus,  BL  =  l,  BL'  =  lf. 
These  magnitudes  are  to  be  reckoned  positive  or  negative  according  as 
they  are  measured  along  the  ray  in  the  same  direction  as  the  light 
goes  or  in  the  opposite  direction. 

If  D  designates  the  foot  of  the  perpendicular  let  fall  from  the  inci- 
dence-point B  on  the  axis,  the  magnitude  DB  =  h  is  called  the  "in- 
cidence-height" of  the  ray,  and  is  reckoned  po.sitive  or  negative  accord- 
ing as  the  point  B  lies  above  or  below  the  axis. 

The  perpendicular  to  the  optical  axis  erected  at  the  centre  C  of 
the  spherical  and  refracting  surf  ace  will  be  called  the  "central  perpen- 
dicular", and  the  intercepts  CH,  CHf  of  the  incident  and  refracted 
rays  on  the  central  perpendicular  will  be  denoted  by  the  symbols  6,  &'; 


296  Geometrical  Optics,  Chapter  IX.  [  §  210. 

thus,  CH  =  b,  CH'  =  b'.  This  intercept  b  is  to  be  reckoned  positive 
or  negative  according  as  the  point  H  lies  above  or  below  the  optical 
axis.  And  a  perfectly  similar  rule  obtains  with  regard  to  the  sign  of  b'. 

2.  Notation  of  the  Angular  Magnitudes. 

The  angles  of  incidence  and  refraction  are  denoted  by  the  symbols 
a  and  a! ';  thus,  in  the  diagram,  Z  CBL  =  a,  Z  CBL'  =  a'.  These 
angles  are  the  acute  angles  through  which  the  radius  CB  must  be 
rotated  around  the  point  B  in  order  to  come  into  coincidence  with  the 
straight  lines  to  which  the  incident  and  refracted  rays  belong. 

The  "Slope"  of  the  incident  ray,  or  its  inclination  to  the  axis,  is 
denoted  by  the  symbol  0;  and,  similarly,  the  symbol  6'  is  used  to 
denote  the  "slope"  of  the  corresponding  refracted  ray.  Thus, 

/.ALB  =  6,     AAL'B  =  B'. 

These  are  the  acute  angles  through  which  the  axis  must  be  turned 
around  the  points  L,  Z/,  in  order  to  be  brought  into  coincidence  with 
the  straight  lines  to  which  the  incident  and  refracted  rays,  respectively, 
belong.  Moreover,  since 

tan  B  =  -  h/DL,     tan  0'  =  -  h/DL', 

the  signs  of  6  and  6'  are  the  same  as  the  signs  of  —  h/v  and  —  h/v', 
respectively. 

The  acute  angle  through  which  the  radius  QB  drawn  to  the  inci- 
dence-point B  has  to  be  turned  around  C  in  order  for  it  to  coincide 
with  CA  will  be  denoted  by  the  symbol  (p\  thus,  Z  BCA  =  <p. 

210.  We  proceed  now  to  remark  a  number  of  useful  relations  be- 
tween the  magnitudes  denoted  by  the  symbols  »,  /,  &,  h,  a,  6  and  <p. 

The  position  of  a  straight  line  is  determined  so  soon  as  we  know 
the  positions  of  two  points  on  the  line  or  the  positions  of  one  point 
together  with  the  direction  of  the  line.  The  equation  of  a  straight 
line  lying  in  a  given  plane — for  example,  the  plane  of  a  Principal 
Section  of  the  spherical  surface — involves  at  most  two  arbitrary  con- 
stants or  parameters;  and  to  each  set  of  values  of  any  such  pair  of 
parameters  there  corresponds  a  perfectly  definite  straight  line  of  the 
given  plane. 

Thus,  for  example,  the  position  (but  not  the  direction)  of  the  in- 
cident ray  LB  lying  in  the  plane  of  the  Principal  Section  will  be 
completely  determined  provided  we  know  the  values,  say,  of  the  param- 
eters v,  0,  which  are  called  by  some  writers  the  "ray-co-ordinates". 
Instead  of  using  »,  0,  we  might  also  define  the  position  of  the  ray  by 


§  210.]  Path  of  Ray  Refracted  at  Spherical  Surface.  297 

means  of  various  other  pairs  of  the  magnitudes  denoted  by  the  symbols 
v,,  I,  b,  h,  a,  6  and  <p.  L.  SEIDEL,  for  example,  uses  in  his  system  of 
optical  formulae  ray-parameters  that  are  equivalent  to  h,  6. 

The  relations  between  these  magnitudes  are  obtained  easily  by  an 
inspection  of  the  triangle  LBC.     Evidently, 

a  =  6  +  <p.  (177) 

This  formula  exhibits  the  connection  between  the  angular  magnitudes. 
By  the  so-called  Law  of  Sines,  we  derive  from  the  triangle  LBC 
the  following  formulae: 

/•sin  6  =  —  r-sin  <p,       "} 

—  r-sin  a  =  (v  —  r)  sin  0,  j-  (178) 

(v  —  r)  sin  <p  =  /-sin  ct\  J 

and  by  the  so-called  Law  of  Cosines: 

I2  =  (v  -  r)2  +  r2  +  zr(v  -  r)  cos  <? 


v  —  r)2  =  r2  +  /2  —  2rl-cos  a, 

r2=  (v-  r)2  +  I2  -  2/0  -  r)  cos  0.    „ 


(179) 


Finally,  by  projecting  two  of  the  sides  of  the  triangle  LBC  on  the 
third  side,  we  obtain: 

r  =  /-cos  a  —  (v  —  r)  cos  <p,  "1 

v  —  r  =  I- cos  0  —  r-cos  <p,  (180) 

I  =  r-cos  a+  (v  —  r)  cos  0.  J 

Also,  in  the  right  triangles  CBD,  LBD,  we  have: 

sin  <p  =  hfr,  (181) 

and 

sin  0  =-/*//.  (182) 

Finally,  if  Y  designates  the  foot  of  the  perpendicular  let  fall  from 
the  centre  C  on  the  straight  line  BH,  we  have  evidently: 

CY  =  r-sm  a  =  b-cos  0.  (183) 

By  priming  the  magnitudes  denoted  by  v,  /,  b,  a  and  0  in  the  above 

formulae  (177),  (178),  (179).  (l8o)»  (lSl)'  (l82)  and  (l83)»  we  sha11 
obtain  the  corresponding  relations  for  the  refracted  ray  BL'. 


298  Geometrical  Optics,  Chapter  IX.  [  §  211. 

ART.  62.     TRIGONOMETRIC  COMPUTATION  OF  THE  PATH  OF 
THE  REFRACTED    RAY. 

211.  The  problem  is  as  follows:  Given  the  spherical  refracting 
surface  and  the  values  of  the  indices  of  refraction  n,  n'  of  the  two 
media  separated  by  it,  and  the  position  of  the  incident  ray,  to  deter- 
mine the  position  of  the  corresponding  refracted  ray.  In  other  words, 
being  given  the  constants  denoted  by  n,  n'  and  r,  and  the  co-ordinates 
v,  0  of  the  incident  ray,  we  are  required  to  find  the  co-ordinates  »',  0' 
of  the  refracted  ray. 

By  the  Law  of  Refraction  : 

n  •  sin  a  =  nf  •  sin  a'. 
Moreover,  since 

a.  =  0  +  <p,     a'  =  0'  +  <p, 

we  have  the  invariant-relation: 

a  -  0  =  af  -  0'. 

By  means  of  these  equations  and  the  second  of  equations  (178)  above, 
we  obtain  easily  the  following  system  of  equations  for  calculating  the 
values  of  v'  ',  0': 

sin  a  =  (i  —  v/r)  sin  0,     sin  a'  =  w-sin  a/n', 

r    (184) 

0'=  e  +  af  -a,  v'  =  r(i  -  sin  a'/sin  #').  J 

We  may  also  remark  here  a  number  of  other  useful  relations  between 
the  parameters  of  the  incident  and  refracted  rays.  For  example, 
since  the  incidence-height  has  the  same  value  for  both  rays,  we  have 
the  following  invariant  relation  : 

I-  smO  =  /'-sin!?';  (185) 

and,  since 

sin  a/sin  6  =  —  c/r,     sin  a'  /sin  0'  =  —  c'  fr, 
and,  therefore, 

sin  6'  /sin  6  =  ncjn'c', 
we  obtain  from  (185): 

nc/l  =  ric'll', 
which  may  also  be  written  : 


v 

This  formula  is,  in  fact,  a  mere  transformation  of  the  Optical  Invari- 

ant (§  25) 


§211.] 


Path  of  Ray  Refracted  at  Spherical  Surface. 


299 


for  the  special  case  of  Refraction  at  a  Spherical  Surface.     The  magni- 
tude 


or 


If 
I  = 


Vr 


(,86) 


K 


r-sm 


which  remains  unchanged  as  the  ray  is  refracted  from  one  medium 
into  the  next,  and  which  may  be  called  the  "invariant  of  refraction 
at  a  spherical  surface",  plays  an  important  part  in  ABBE'S  Theory  of 
Spherical  Aberration. 

Note  i.  In  the  special  case  of  Reflexion  at  a  Spherical  Mirror, 
we  have  only  to  put  n'  =  —  n  in  the  above  formulae  (see  §  26).  For 
example,  putting  n'  =  —  n  in  formulae  (184),  we  obtain: 


sin  a  =  sin  0(i  —  v/r), 


«=-«, 

e'  =  e  -  2 


sn 


Reflexion  at 
Spherical  Mirror. 


Note  2.  The  following  formula,  adapted  to  logarithmic  computation, 
is  convenient  as  a  ' 'check"  formula  in  calculating  the  magnitude  v' : 

v'  =  AL'  =  AC  +  CD  +  DL' 
—  r  —  r  -  cos  <p  —  h  -  cot  0' 
=  2r  -  sin2  (p/2  —  r  •  sin  <p  •  cos  0'  /sin  6' 
cos0' 


(<p  <p    cos  6  \    ,     <p 

sm cos  -  •  - — -.  I  sin  -  ; 
2  2    sin  0'  /        2 

so  that,  finally,  we  may  write : 

(\                                     *   I   fif 
6'  +  -  )           2r  •  sin  -  •  cos  — 
2/ 2 2 


»' 
and,  similarly: 


sm 


2r 


•S:n2-COS("+f)_ 


sin0 


sin(9' 


.    <P  a 

2r-sm-  •  cos  — 

sin  e 


300  Geometrical  Optics,  Chapter  IX.  [  §  211. 

Dividing  one  of  these  formulae  by  the  other,  we  obtain : 

,      sin  6  •  cos 

v 

V 


sin  6'  -  cos 

In  the  special  case  of  Refraction  at  a  Plane  Surface,  putting  r  —  oo, 
we  have  <p  =  o,  and  the  above  formula  becomes: 

t/-tan0'  =  p-tan0,  (Refraction  at  Plane  Surface), 

which  may  also  be  easily  derived  directly  (§  52). 

Note  3.  Spherical  Aberration.  The  co-ordinates  v' ',  6'  of  the  re- 
fracted ray  BR'  can  be  found,  as  we  have  shown,  in  terms  of  the  co- 
ordinates v,  0  of  the  incident  ray.  If  in  the  formula 

n(v  —  r)/l  =  n'(vr  —  r)/l' 

we  substitute  for  /,  /'  their  values  as  given  by  the  first  of  formulae  (179), 
it  is  obvious  that  v'  will  thus  be  expressed  as  a  function  of  n,  n' ,  r,  v 
and  (p.  The  magnitudes  n,  n'  and  r  are  constants,  so  that  v'  is,  in 
fact,  a  function  of  the  variables  v  and  <p;  and,  therefore,  if  v  is  kept 
constant,  it  is  obvious  that  we  shall,  generally,  obtain  different  values 
of  vf  by  merely  changing  the  value  of  <p.  This  is  the  analytical  state- 
ment of  the  fact  of  Spherical  Aberration  mentioned  in  §  208. 

The  positions  on  the  axis  of  the  so-called  "aplanatic"  pair  of  points 
Z,  Z'  (§  207)  can  be  found  easily  by  means  of  the  formulae  obtained 
above.  The  condition  that  the  abscissa  v'  corresponding  to  a  certain 
fixed  value  of  v  shall  be  independent  of  the  angle  <p  must  be  imposed 
upon  the  equations.  Since 

nc/l  =  ric'\V, 
and 

I2  =  c2  +  rz  +  2rc-cos  <p, 

2  2 

we  obtain : 

(n'2  -  n2)c2cf2  +  (n'2c'2  -  nzc2)r2  +  2rcc'(»'V  -  nzc)  cos  9  =  o. 

If,  for  a  given  value  of  c,  the  value  of  c'  derived  from  this  equation 
is  to  be  independent  of  <p,  we  must  have: 


c  = 
n 


§  211.]  Path  of  Ray  Refracted  at  Spherical  Surface.  301 

which  shows  that  for  this  particular  pair  of  values  c,  c'  must  have  the 
same  sign;  that  is,  the  points  Z,  Z'  must  lie  on  the  same  side  of  the 
centre  C.  If  in  the  equation  above  we  substitute  this  special  value 
of  c',  we  obtain 

/2 


This  equation  gives  two  values  of  c,  of  which  only  the  value 


is  admissible  here  where  we  have  to  do  with  optical  rays  as  distin- 
guished from  mere  geometrical  rays.  (The  value  c  =  —  n'rjn  cor- 
responds to  the  other  intersection  of  the  ray  with  the  auxiliary  spheri- 
cal surface  r  in  Figs.  114  and  115.)  Thus,  we  find: 

v  =  r  +  n'rjn,     v'  =  r  -f-  nrjn' 

for  the  abscissae  A  Z,  A  Zf  of  the  pair  of  aplanatic  points  of  a  spherical 
refracting  surface;  in  agreement  with  the  results  of  §  207. 

A  characteristic  property  of  the  aplanatic  points  of  a  single  spheri- 
cal refracting  surface,  which  was  also  remarked  in  §  207,  may  be  stated 
as  follows:  If  9,  6'  denote  the  slopes  of  the  incident  and  refracted 
rays  BZ,  BZ',  then 

sin  6  /sin  6'  —  njn'\ 

xthat  is,  the  ratio  of  the  sines  of  the  "slopes-angles  is  independent  of 
the  magnitude  of  the  angle  of  incidence,  and  constant,  therefore,  for 
all  pairs  of  corresponding  incident  and  refracted  rays.  If  Y  denotes 
the  value  of  the  Lateral  Magnification  by  means  of  Paraxial  Rays  for 
the  pair  of  conjugate  points  Z,  Z',  we  shall  find  that: 


and,  hence,  the  relation  obtained  above  may  be  written: 

sin0/sin0'  =  n'Y/n. 

Expressed  in  this  form,  this  relation,  which  we  have  obtained  for  the 
aplanatic  points  of  a  single  spherical  refracting  surface,  represents 
a  very  important  general  law  of  Optics  known  as  the  Sine-  Condition 
(Art.  86),  which  will  be  fully  considered  in  a  subsequent  chapter. 
Note  4.  If  the  position  of  the  ray  is  defined  by  means  of  its  slope- 


302  Geometrical  Optics,  Chapter  IX.  [  §  212. 

angle  0  and  its  intercept  b  on  the  central  perpendicular,  then  by 
formula  (183): 

r  •  sin  a  =  b-  cos  0,     r  •  sin  a   =  b'  •  cos  B' ; 

and,  hence,  since 

w-sin  a  =  w'-sin  a', 

we  obtain  the  following  invariant-relation: 

n  -  b  -  cos  6  =  n'-b'-  cos  0'. 

Thus,  being  given  the  parameters  b,  B  of  the  incident  ray,  we  can  find 
the  parameters  b' ,  B'  of  the  refracted  ray  by  means  of  the  following 
system  of  equations: 


b  •  cos  B                         n-sina 
sin  a  =  -     ,         sm  a   = ~f —  , 


n    cos  8  . 
---—  b. 
n  cos0' 


(187) 


Since  CH  =LC-tan  0,  the  connection  between  the  intercept  AL  =  v 
on  the  optical  axis  and  the  intercept  CH  =  b  on  the  central  perpen- 
dicular is  given  by  the  formula  : 

b  =  (r  -  v)  tan  0. 

ART.  63.     FORMULA  FOR    FINDING   THE    POINT   OF   INTERSECTION   AND 

THE  INCLINATION  TO  EACH  OTHER  OF  A  PAIR   OF  REFRACTED 

RAYS  LYING  IN  THE   PLANE   OF  A  PRINCIPAL   SECTION 

OF  THE  SPHERICAL  REFRACTING  SURFACE. 

212.  Let  one  of  the  incident  rays,  distinguished  as  the  chief  of  the 
two,  cross  the  optical  axis  at  the  point  L  (Fig.  121);  which,  when  the 
ray  is  the  chief  of  a  bundle  of  incident  rays,  will  coincide  with  the 
position  of  the  centre  of  the  "stop",  or  circular  diaphragm,  which  is 
used  to  limit  the  bundle  of  object-rays  that  are  permitted  to  pass 
through  the  optical  system;  and  let  the  incidence-point  of  the  chief 
ray  be  designated  by  the  letter  B.  The  other  ray  (which  we  may 
call  the  "secondary  "ray)  crosses  the  optical  axis  at  the  point  L,  and 
meets  the  spherical  surface  at  the  point  B.  The  positions  of  both  of 
these  rays  are  supposed  to  be  known,  so  that  we  may  consider  that  we 
know  their  "slopes", 


and  their  intercepts  on  the  optical  axis, 

v  =  AL,     v  =  AL] 


§  212.]  Path  of  Ray  Refracted  at  Spherical  Surface.  303 

so  that  we  also  know  (or  can  find)  the  angles  of  incidence, 
a=/.CBT,     a= 


the  point  of  intersection  of  the  secondary  ray  with  the  chief  ray  being 
designated  in  the  diagram  by  the  letter  T.  The  magnitudes 

BT  =  t,      ^BTB  =  \ 

may  be  regarded  as  the  co-ordinates  of  the  secondary  ray  with  respect 
to  the  chief  ray.  This  intercept  t  on  the  chief  ray  is  measured  always 
from  the  incidence-point  B  as  origin,  and  is  to  be  reckoned  positive 


FIG.  121. 

Figure  represents  a  pair  of  rays,  lying  in  the  plane  of  a  principal  section  of  a  spherical  refracting 
surface,  and  incident  on  this  surface  at  the  points  designated  by  B  and  B.  These  rays  cross  the 
optical  axis  at  the  points  designated  by  L,  and  L,  and  intersect  each  other  at  the  point  designated 
by  T.  The  refracted  rays  are  not  shown. 


AL  =  v,    AL  =  v,    AC=r,    BT=  t, 

L  BCB  =x,     £  ALB  =  9,     /  ALB  =  9,     Z  BTB  =  A. 

or  negative  according  as  the  light  travels  along  the  straight  line  BT 
in  the  direction  from  B  towards  T  or  in  the  opposite  direction.  The 
"aperture-angle"  X  is  defined  as  the  acute  angle  through  which  the 
chief  ray  BT  must  be  turned  around  the  point  T  in  order  to  bring  it 
into  coincidence  with  the  secondary  ray  BT. 
Putting 


we  have: 

<P  =  $  +  X,  (188) 

where  \  —  £B  CB  denotes  the  increase  of  the  central  angle  <(>. 
From  the  figure  we  have  evidently: 

a  —  X=  a  +  x-  (189) 


304  Geometrical  Optics,  Chapter  IX.  [  §  213. 

From  the  centre  C  draw  CY  perpendicular  to  the  straight  line  BT 
at  Y.  The  orthogonal  projection  of  the  radius  CB  on  the  straight 
line  CY  is  equal  to  the  sum  of  the  orthogonal  projections  on  CY  of 
the  line-segments  CB  and  BT\  and,  since  these  projections  are  equal 
to  r-sina,  r-sin(a  +  X)  and  —  /-sinX,  respectively,  we  obtain  the 
relation  : 

r-sin  a  =  r-sin  (a  -f-  X)  —  /-sin  X, 
or 

t  •  sin  X 
---  =  sin  (a  +  X)  —  sin  a.  (190) 

If  in  formulae  (189)  and  (190)  we  prime  the  symbols  /,  X,  a  and  a, 
we  shall  obtain  the  formulae  for  the  corresponding  pair  of  refracted  rays. 

Knowing,  therefore,  the  positions  of  the  pair  of  incident  rays,  and 
being  given  the  values  of  the  magnitudes  denoted  by  /,  X,  we  can 
find  the  values  of  the  magnitudes  denoted  by  t'  ,  X'.  Thus,  since 

a  —  X  —  a  =  a'  —  X'  —  a'  =  x, 
and 

/  •  sin  X  N  /'  •  sin  X'         .  A 

=  sin  (a  +  X)  —  sin  a,         ---  =  sin  (a  +  X  )  —  sin  a  , 


and,  also,  since 

a!  +  a'  +  X7 

/    /    i    >  /\  •         /        COS  — 

sin  (a  +  X  )  —  sin  a  __  2 

sin  (a  +  X)  —  sin  a  a  +  a  +  X    ' 

cos  — 

2 

we  derive  the  following  formulae  : 

X'=X+(a-o/)  -  (a  +  «'),! 
a'  +  a'  +  V 


t'      sin  X 


cos 


sin  X'          a  +  a  +  X 
cos  — 


(191) 


CASE  II.    WHEN  THE  PATH  OF  THE  RAY  DOES  NOT  LIE  IN  A  PRINCIPAL 
SECTION  OF  THE  SPHERICAL  REFRACTING  SURFACE. 

ART.  64.     PARAMETERS  OF  OBLIQUE  RAY. 

213.  The  equation  of  a  straight  line  in  space  involves  as  many  as 
four  arbitrary  constants,  and  the  forms  of  the  refraction-formulae  which 
we  shall  obtain  will  depend  on  how  these  ray-parameters  are  chosen. 

Let  us  take  the  centre  C  of  the  spherical  refracting  surface  as  the 


§  214.]  Path  of  Ray  Refracted  at  Spherical  Surface.  305 

origin  of  a  system  of  rectangular  co-ordinates.  Naturally,  also,  we 
shall  take  the  optical  axis  itself  as  the  #-axis.  The  plane  of  a  prin- 
cipal section  of  the  spherical  surface  may  be  conveniently  selected  as 
the  xy- plane ;  nor  will  it  at  all  affect  the  generality  of  the  following 
treatment  if  for  this  plane  we  take  that  meridian  section  of  the  spher- 
ical surface  which  contains  also  the  object-point.  The  plane  of  the 
principal  section,  which  is  perpendicular  to  the  :ry-plane,  will  then  be 
the  xz-plane,  and  a  transversal  plane  at  right  angles  to  the  optical 
axis  will  be  the  jz-plane.  For  convenience,  we  may  suppose  that  the 
axis  of  y  is  vertical,  and  that  the  axes  of  x  and  z  are  horizontal. 

The  letters  G,  H  and  /  will  be  used  to  designate  the  points  where 
the  incident  ray,  prolonged  if  necessary,  crosses  the  xy-t  yz-  and  xz- 
planes,  respectively;  and  the  rectangular  co-ordinates  of  these  points 
will  be  denoted  by 

xgj  yg,  o;     o,  yh,  zhJ     and     x{,  o,  z., 
respectively. 

In  the  following  we  shall  explain  the  methods  of  A.  KERBER  and 
L.  SEIDEL  of  calculating  the  path  of  a  ray  refracted  obliquely  at  a 
spherical  surface. 

214.     Method  of  A.  Kerber. 

In  the  calculation-system  of  A.  KERBER/  the  position  of  the  ray 
is  determined  by  the  co-ordinates  of  the  points  G  and  /,  where  the 
ray  crosses  the  vertical  plane  of  the  principal  section  (xy-plane)  and 
the  horizontal  meridian  plane  (xz-plane).  In  the  figure  (Fig.  122) 
the  spherical  triangle  AAgA{  represents  a  piece  of  the  spherical  re- 
fracting surface.  The  point  A,  where  the  optical  axis  crosses  this 
surface,  is  the  vertex  of  the  surface ;  AAg  C  is  the  plane  of  the  principal 
section,  and  AA{C  is  the  meridian  section  perpendicular  to  the  prin- 
cipal section.  Let 

^ACAg  =  <pg,     ^ACA,=  <f>{. 

These  angles  are  precisely  defined  by  the  following  relations: 

'V  £* 

tan  <p   =  -  -9 ,     tan  ^  =  -  -' .  (192) 

xg  %i 

Also,  regarding  the  radius  AgC  as  a  secondary  axis  of  the  spherical 
surface,  let  us  denote  the  abscissa  of  the  point  G,  with  respect  to  Ag 
as  origin,  by  vg\  and,  similarly,  regarding  the  radius  A{C  as  another 
secondary  axis,  we  shall  denote  the  abscissa  of  the  point  /,  with  respect 
to  Ai  as  origin,  by  vt\  thus,  vg  =  AgG,  vi  =  AJ.  From  the  figure, 

1  A.  KERBER:  Beitraege  zur  Dioptrik,  Heft  II  (Leipzig,  GUSTAV  FOCK,  1896),  pages  5-8. 
21 


306 

we  obtain: 
and,  since, 

we  have: 


Geometrical  Optics,  Chapter  IX. 
xg=   CG  -  cos  tpgt     xi  =  CI  -  cos 
CG  =  vg  —  r,     CI  =  v{  -  r, 

xa  x: 


( §  214. 


v   —  r  = 


v  —  r  = 


(193) 
cos  <pg  cos  <pt 

The  projection  of  the  incident  ray  in  the  plane  of  the  principal 
section  (jcy-plane)  makes  with  the  optical  axis  an  angle  e,  and  with 


FIG.  122. 


KERBER'S  METHOD  OF  DEALING  WITH  THE  OBLIQUE  RAY.  The  plane  of  the 
represents  a  principal  section  of  a  spherical  refracting  surface,  centre  at  C,  and  optical  axis 
coinciding  with  the  .ar-axis  of  co-ordinates.  A  ray,  whose  path  does  not  lie  in  the  plane  of  the 
principal  section,  is  incident  on  the  spherical  surface  at  the  point  B.  This  ray  crosses  the  ^y-plane 
at  the  point  designated  by  G  and  the  jr-jr-plane  at  the  point  designated  by  7.  The  spherical  triangle 
AAgAi  is  formed  by  the  intersections  of  the  vertical  .arj'-plane,  the  horizontal  -r^r-plane  and  the 
plane  of  incidence  with  the  spherical  refracting  surface. 

LACA,***,.     Z.ACAi  =  fa,     Z.  AgGB  =  9,,     LAiIB  =  9it     Z.  GBC  =  a,    AgG  =  vg,    AJ=  vi. 

the  ray  itself  an  angle  6;  these  angles  being  exactly  defined  by  the 
following  formulae: 


tan  €  = 


xt  — 


Z{  -  COS  € 

tan  8  =  - 


(194) 


§  215.]  Path  of  Ray  Refracted  at  Spherical  Surface.  307 

Moreover,  let  B  designate  the  point  where  the  ray  meets  the  spheri- 
cal refracting  surface,  and  let  us  put 

Z.AgGB  =  8g,     Z.AJB  =  0f. 
The  angle  Bg  may  be  determined  from  the  following  relation : 

cos  Bg  =  cos  (e  -  <pg)  •  cos  d,  (195) 

which  may  easily  be  derived  from  the  figure ;  and  the  angle  Ot  may  be 
determined  in  terms  of  Bg  by  means  of  the  formula: 

sin04  =^—  -smOg,  (196) 

v{  —  r 

which  may  also  be  derived  without  difficulty. 

The  plane  AgAiC  contains  the  incident  ray  GI  and  the  incidence- 
normal  B  C,  so  that  this  plane  is  the  plane  of  incidence.  The  radii 
AgC,  AtC  both  lie  in  this  plane,  as  do  also  the  line-segments  denoted 
by  vg1  vt  and  the  angles  denoted  by  Og1  Ot;  and,  consequently,  regarding 
AgC  and  A{C  each  as  axes  of  the  spherical  surface,  we  have  evidently 
the  following  relations  exactly  similar  to  the  relations  expressed  by 
equation  (177)  and  the  second  of  equations  (178): 

a  =  Bg  +  <pg  =  9i  +  <ft,  (197) 

and 

—  r  •  sin  a.  =  (vg  —  r)  sin  Bg  =  (vt  —  r)  sin  Oit  (198) 

where  a  denotes  the  angle  of  incidence. 

If  in  the  figure  the  letters  G  and  /  are  primed,  the  diagram  will 
answer  to  show  the  corresponding  case  of  a  ray  refracted. at  a  spherical 
surface,  and  by  priming  all  the  symbols  x,  y,  2,  v,  6,  a,  e  and  d  in  the 
formulae  (192)  to  (198)  above,  we  shall  obtain  the  corresponding  rela- 
tions between  the  parameters  of  the  refracted  ray. 

215.    Method  of  L.  Seidel. 

Instead  of  determining  the  position  of  the  ray  by  its  points  of  inter- 
section with  two  selected  planes,  L  SEIDEL  1  makes  use  of  only  one 
such  point,  and,  in  place  of  the  co-ordinates  of  a  second  point,  employs 
two  angular  parameters  to  define  the  direction  of  the  ray.  The  point 
of  the  ray  which  he  selects  is  the  point  designated  by  H  (Fig.  123) 

*L.  SEIDEL:  Trigonometrische  Formeln  fur  den  allgemeinsten  Fall  der  Brechung  des 
Lichtes  an  centrirten  sphaerischen  Flaechen:  Sitzungsber.  der  math.-phys.  Cl.  der  kgl.  bayr. 
Akad.  der  Wissenschaften,  vom  10.  Nov.  1866.  Reprinted  in  Beilage  III  of  STEINHEIL  & 
VOIT'S  Handbuch  der  angewandten  Optik,  Bd.  I  (Leipzig,  B.  G.  TEUBNER,  1891),  pager 
257-270. 


308 


Geometrical  Optics,  Chapter  IX. 


[§215. 


where  the  ray  crosses  the  transversal  (or  yz-)  plane.  Moreover,  in- 
stead of  using  the  rectangular  co-ordinates  (yh,  zh)  of  this  point,  he 
introduces  a  system  of  polar  co-ordinates  (p,  TT)  in  the  yz-plane.  Em- 
ploying other  symbols  than  those  used  by  SEIDEL  himself,  we  shall 
write : 

p  =  CH,     TT  =  Z  HCy, 

which  magnitudes  are  connected  with  the  rectangular  co-ordinates  of 
H  by  the  following  relations : 

yh  =  p-cos  TT,     zh  =  p-sin  IT.  (199) 

Both  the  radius-vector  p  and  the  polar  angle  TT  are  to  be  considered  as 
always  positive  in  sign.  The  angle  TT,  which  may  thus  have  any  value 


METHOD  OF  I,.  SEIDEL.  The  straight  line  BH  represents  a  ray  incident  obliquely  at  the  point  B 
on  a  spherical  refracting  surface,  whose  centre  is  at  the  point  designated  by  C.  The  optical  axis 
coincides  with  the  x-axis  of  co-ordinates,  and  the  plane  of  the  paper  is  the  plane  of  a  principal 
section  (^v-plane) ;  An  being  the  section  of  the  spherical  surface  made  by  this  plane.  CB  is  the 
incidence-normal,  and  ACB  is  the  plane  of  incidence.  The  ray  BH  crosses  the  ^try-plane  at  the 
point  designated  by  G,  and  the  ^-plane  at  the  point  designated  by  H.  The  polar  co-ordinates  of 
the  point  Hare  p  =  CH,  IT  =  Z.  HCy.  The  angle  at  B  is  the  angle  of  incidence  a.  The  acute  angle 
made  by  the  ray  with  the  .r-axis  is  the  angle  denoted  by  r ;  and  the  angle  made  by  the  projection 
of  the  ray  on  theyz-plane  with  the  positive  direction  of  the  j'-axis  is  the  angle  denoted  by  ^. 

comprised  between  o°  and  360°,  may  be  defined  as  the  angle  through 
which  CH  has  to  be  turned  about  C,  always  in  the  sense  of  positive 
rotation,  in  order  that  it  may  come  into  coincidence  with  the  positive 
direction  of  the  ;y-axis. 


§  215.]  Path  of  Ray  Refracted  at  Spherical  Surface.  309 

One  of  the  two  angular  magnitudes  that  define  the  direction  of  the 
ray  is  the  acute  angle  (T)  between  the  direction  of  the  ray  and  the 
positive  direction  of  the  jc-axis;  this  angle  being  reckoned  always  as 
positive. 

The  other  angular  magnitude  selected  for  this  purpose  by  L.  SEIDEL 
is  the  angle  (\f/)  made  with  the  positive  direction  of  the  ^-axis  by  the 
projection  of  the  ray  on  the  transversal  (or  yz-)  plane.  This  angle, 
likewise,  is  always  reckoned  as  positive,  but  it  may  have  any  value 
comprised  between  o°  and  360°. 

If  the  direction-cosines  of  the  straight  line  HI  are  denoted  by  a, 
0,  T,  then 

-  =  _  £  =      T 

*i      y*.    *<-**' 

and,  since 

tan  ^  =  7//3 

(as  may  be  easily  verified) ,  we  obtain : 

&      cj 

tan  \J/  =  — l ,  (200) 

whereby  the  angle  \j/  is  precisely  defined. 
Moreover,  since 

a2  -\-  /32  +  72  =  i ,     and     a  =  cos  T, 

we  find  (taking  the  minus  sign,  which  is  in  agreement  with  the  defi- 
nitions above) : 

]8  =  —  sin  r-cos  ^; 
and,  hence: 

tan  T  =  — -*— r,  (201) 

x^cos  \f/ 

or 


which  is,  therefore,  the  definition-equation  of  the  angle  r. 

An  auxiliary  angle  (/i)  is  also  employed  in  the  calculation-scheme 
of  L.  SEIDEL.  Let  B  designate  the  point  where  the  ray  meets  the 
spherical  surface;  in  the  triangle  BHC,  the  angle  at  H,  but  not 
necessarily  the  interior  angle,  is  the  angle  denoted  by  ju.  This  angle, 
which  is  also  reckoned  as  positive,  may  have  any  value  comprised 
between  o°  and  180°,  and  is  defined  exactly  by  the  following  formula: 

cos  IJL  =  —  sin  r-cos  (\f/  —  TT),  (202) 


310  Geometrical  Optics,  Chapter  IX.  [  §  216. 

a  relation  which  may  easily  be  verified  from  the  above  definitions  of 
the  angles  denoted  by  TT,  r  and  \f/. 

From  the  triangle  B  H  C  we  derive  also  the  following  formula  con- 
necting the  angle  of  incidence  a  at  B  and  the  auxiliary  angle  ju  at  H: 

r-sin  a  =  p-sin  /*;  (203) 

wherein  it  should  be  noted  that,  according  to  this  formula,  since  by 
definition  both  p  and  sin  /z  are  positive  magnitudes,  and  the  angle  a 
is  an  acute  angle,  the  sign  of  the  angle  a  must  be  reckoned  always  as  the 
same  as  the  sign  of  the  radius  r.1 

The  point  where  the  refracted  ray  crosses  the  transversal  ^z-plane 
is  designated,  similarly,  by  H' ';  and  if  the  symbols  x,  y,  z,  p,  IT,  r,  $,  n 
and  a  in  formulae  (199)  to  (203)  above  are  primed,  we  shall  obtain 
at  once  the  relations  between  the  corresponding  magnitudes  which 
relate  to  the  refracted  ray. 

ART.  65.     TRIGONOMETRIC  COMPUTATION  OF  PATH  OF  RAY  REFRACTED 
OBLIQUELY  AT  A  SPHERICAL  SURFACE. 

216.    The  Refraction-Formulae  of  A.  Kerber. 

The  problem  is  as  follows :  Being  given  the  rectangular  co-ordinates 
(xg1  yg)  and  (xit  zt.)  of  the  points  G  and  I  where  the  incident  ray  crosses 
the  xy-  and  xz-planes,  respectively,  to  determine  the  co-ordinates 
(x'g>  y'g)  and  (x'n  z'i)  °f  tne  corresponding  points  Gf  and  /'  where  the 
refracted  ray  crosses  these  same  planes. 

By  the  Law  of  Refraction,  we  have: 

w-sin  a  =  w'-sin  a:'; 
and,  moreover,  since 

a  =  eg  +  <p8^ei  +  ft,    a  =  e'g  +  <pg  =  e\  +  <?» 

we  have : 

a  -  eg  =  d  -  o'g,    a  -  e{  =  af  -  e\. 

By  means  of  these  formulae  and  the  formulae  (192)  to  (198),  we  obtain 
A.  KERBER's2  System  of  Refraction- Formula,  as  follows: 

1  This  is  practically  equivalent  to  the  method  used  by  B.  WANACH  in  a  paper  entitled 
Ueber  L.  v.  SEIDEL'S  Formeln  zur  Durchrechnung  von  Strahlen  durch  ein  zentriertes  Lin- 

sensystem,  nebst  Anwendung  auf  photo  gr  a  phis  che  Objective,  published  in  Zeitschrift  fur  In- 
strumentenkunde,  xx.  (1900),  pp.  162-171.  In  SEIDEL'S  formulae,  as  originally  publishedr 
the  symbol  R  is  used  to  denote  the  absolute  value  of  the  radius  of  the  refracting  surface, 
so  that  SEIDEL  has  to  employ  the  double  sign  in  order  to  include  the  cases  of  both  convex 
and  concave  surfaces.  SEIDEL  adopted  this  method  by  preference,  as  being,  in  his  opinion, 
practically  the  most  convenient. 

2  A.  KERBER:  Beitraege  zur  Dioptrik.     Zweites  Heft.  (Leipzig,  GUSTAV  FOCK,  1896), 
pages  5-8. 


§217.] 


Path  of  Ray  Refracted  at  Spherical  Surface, 
tan  <pQ  =  -  yjxg,     tan  ^  =  —  zi\xi ; 


311 


tan  €  =  — ^f  —  ; 

xc 


tan  5  = 


Zi  •  COS  € 


r  -  v,  =  - 


COS 

r  — 


cos  Og  =  cos  (e  —  <pg)  •  cos  5,     sin  0t-  =  •— —  sin  0, 


r  —  vg  . 
sin  a  = sin  0_ ; 


r-sm  a 


sin 


w    . 
—  sin  a ; 


r-sm  a 


sin0 


-  (r-  v'g)  •  cos  <pg,    x\  =  -  (r  -  »'.)  •  cos 

si  =  -  *i  - tan  ^  • 


(204) 


xa 


217.  In  the  special  case  of  a  Plane  Refracting  Surface,  the  centre 
C  is  the  infinitely  distant  point  of  the  optical  axis,  and,  hence, 
the  origin  of  co-ordinates  will  have  to  be  shifted  from  C  to  the  point  A 
where  the  optical  axis  meets  the  refracting  plane,  which  is  effected 
very  simply  by  writing  x  —  r  in  place  of  x.  If  we  do  this,  and  then 
put  r  =  oo ,  KERBER'S  Formulae  for  a  Plane  Refracting  Surface  will  be 
found,  as  follows: 


<Pi 


tanc  = 


x,  — 


tan  5 


zi  -  cos  c 


vn  =  xn.    vt  =  Xii 

g  ff'  »  f ' 

cos  Bg  =  cos  e  •  cos  5 ; 


sin  6'g  =  —,  sin 
tan  B 


=  ^ 


ten*;1 


Xg=Vg',  X 

yg  ~  ^» 


V.- 


tan  0t. 
'tan  0V 


(205) 


312  Geometrical  Optics,  Chapter  IX.  [  §  218. 

218.     In  case  the  angle  Bg  is  very  small,  the  determination  of  this 
angle  by  means  of  the  formula 

cos  Qg  =  cos  (e  —  <pg)  -  cos  6 

is  not  satisfactory,  and  a  greater  numerical  accuracy  will  be,  possible 
by  determining,  first,  the  value  of  the  angle  (3  between  the  plane  of 
incidence  and  the  vertical  plane  of  the  Principal  Section  by  means  of 
the  following  formula  :  l 

tan  5 

tan  |8  =  -7—7  --  x  ;  (206) 

sin  (e  -  <pg) 


whence  we  can  find  afterwards: 

sin  d 


(207) 


In  connection  with  KERBER'S  Refraction-Formulae,  the  following 
suggestion,  also  due  to  Messrs.  KOENIG  and  VON  ROHR,Z  is  worthy  of 
remark: 

By  taking  as  the  ray-parameters  the  co-ordinates  xg,  yg  and  the 
angular  magnitudes  denoted  by  5  and  e,  the  calculation  of  all  of  the 
magnitudes  denoted  above  by  symbols  with  the  subscript  i  can  be 
entirely  avoided.  Since,  by  (207),  we  have: 

sin  5  sin  5' 

we  obtain: 

sin  6'  =  ~r-j  sin  5,  (208) 

whereby  we  can  determine  the  angle  5';  and  the  value  of  the  angle  e' 
may  be  found  by  the  formula: 

COS    Og 

cos  (e  —  <pg)  = —, , 

or  by  the  formula: 

tan  6' 
sin  (e'  -  <f>a)  = 


1  This  suggestion  is  found  in  Die  Theorie  der  optischen  Instrumente  (Berlin,  JULIUS 
SPRINGER,  1904),  Bd.  I,  II  Kapitel,  "Die  Durchrechnungsf ormeln " :  von  A.  KOENIG  und 
M.  VON  ROHR,  p.  65. 

2  Same  reference  as  preceding. 


§  219.]  Path  of  Ray  Refracted  at  Spherical  Surface.  313 

219.    The  Refraction-Formulae  of  L.  Seidel.1 

Here  the  problem  is  as  follows :  Being  given  the  angular  magnitudes 
(r,  i/O,  which  define  the  direction  of  the  incident  ray,  and  the  polar 
co-ordinates  (p,  IT)  of  the  point  H  where  this  ray  crosses  the  ;ys-plane, 
to  find  the  corresponding  parameters  (rr ,  \f/f)  and  (//,  TT')  of  the  re- 
fracted ray. 

Since  the  plane  of  the  triangle  BHC  contains  the  incident  ray  BH 
and  the  incidence-normal  BC,  this  is  the  plane  of  incidence,  which 
likewise,  therefore,  contains  the  refracted  ray  BH'.  That  is,  the  two 
planes  BHC  and  BH' C  coincide,  and,  consequently,  their  lines  of 
intersection  with  the  yz- plane  coincide  also.  Hence,  the  three  points 
C,  H  and  H'  all  lie  on  one  and  the  same  straight  line;  accordingly, 
the  radii  vectores  CH,  CHr  have  the  same  (or  opposite)  directions, 
so  that  the  polar  angles  TT,  TT'  are  either  equal  or  differ  by  180°.  In  the 
case  of  a  refracting  surface,  we  shall  have: 

x'  =  x; 
and  for  a  reflecting  surface : 

TT'  =  180°  +  TT. 
By  formula  (203),  we  have: 

r-sin  a  =  p-s'm  n,     r-sin  a.'  =  p'-sm  //, 

where  /*,  y!  are  the  two  auxiliary  angles  at  the  vertices  H,  H'  of  the 
triangles  BHC,  BH'C\  and  hence,  by  the  Law  of  Refraction,  we 
derive  the  invariant  relation: 

n  -  p  •  sin  fj,  —  n'-p'-sin  /*'.  (209) 

Moreover,  since  the  angle  at  C  is  common  to  these  two  triangles,  we 
obtain  also  another  invariant  relation  as  follows: 

/*  +  «  =  /*'  +  «'.  (210) 

By  means  of  the  above  formulae,  the  position  of  the  point  H'  may  be 
determined. 

Still  another  invariant  relation,  depending  on  the  fact  that  the 
plane  of  incidence  and  the  plane  determined  by  the  optical  axis  and 
the  radius  vector  CH  coincide  with  the  plane  of  refraction  and  the 
plane  determined  by  the  optical  axis  and  the  radius  vector  CH' ,  re- 

1  L.  v.  SEIDEL:  Trigonometrische  Formeln  fiir  den  allgemeinsten  Fall  der  Brechung 
des  Lichtes  an  centrierten  sphaerischen  Flaechen:  Sitzungsber.  der  math.-phys.  Cl.  der  kgl> 
bayr.  Akad.  der  Wissenschaften,  vom  10.  Nov.  1866.  Reprinted  in  Beilage  III  of  Bd. 
I  of  STEINHEIL  &  VOIT'S  Handbuch  der  angewandten  Optik  (Leipzig,  B.  G.  TEUBNER,  1891). 


314  Geometrical  Optics,  Chapter  IX.  [  §  219. 

spectively,  which  may  also  be  easily  derived,  is  the  following: 
sin  r'-sin  (\j/f  —  TT)       sin  r-sin  (\f/  —  TT) 


sm 


sin  fj, 


Moreover,  the  following  relation,  also,  is  obvious: 


(211) 


COS   T 


cos  r 


sm  //      sm  fj, 

but  this  formula,  which  is  convenient  by  reason  of  its  simplicity,  is  not 
a  very  practical  formula  for  numerical  calculation  in  case  the  angles 
T,  r'  are  small,  as,  in  fact,  they  usually  are.  But  if  we  combine  this 
formula  with  (211),  we  obtain: 


tan  r'  •  sin  (^'  —  TT)  =  tan  r  •  sin  (i/>  —  TT)  ; 


(212) 


whereby  we  can  find  the  tangent  of  the  angle  T'. 

Finally,  arranging  the  above  formulae  in  the  order  in  which  they  are 
used,  we  have  L.  SEIDEL'S  Calculation- Scheme  for  determining  the 
refracted  ray  corresponding  to  a  ray  incident  obliquely  on  a  spherical 
refracting  surface,  as  follows : 

(i)  Determination  of  the  Position  of  H'  by  means  of  its  Polar  Co- 
ordinates (pf,  IT'}: 

cos  /z  =  —  sin  r-cos  (\f/  —  TT), 


sin  a  = 


p-sin  fi. 
r 

n-sin  a. 


f  _       sn  a  n  sn 

sin  /*'  n'  sin 


7T     =   7T. 


(213) 


Note.  —  As  we  shall  have  to  calculate  below  the  quotient  sin  ju'/sin  M, 
it  is  worth  while  to  compute  the  value  of  p'  by  means  of  each  of  the 
two  formulae  above;  as  this  will  afford  us  some  way  of  checking  the 
values  obtained  for  the  angles  a,  a'. 


§  220.]  Path  of  Ray  Refracted  at  Spherical  Surface.  315 

(2)  Determination  of  the  Direction  (r ',  f)   of  the  Refracted  Ray' 


'  —  TT)  =  -: — '—  -sin  r •  sin  (\l/  —  TT) , 
sm  /* 


tan  (x  -  ^')  = 


sin  r'  •  sin  (\f/r  —  TT) 


cos  // 


sin  (i£  —  TT) 

tan  r'  =  tan  r  —777-  -4  . 
sm  (^   —  TT) 


(214) 


.  —  The  second  of  these  formulae  is  obtained  by  combining  for- 
mula (212)  with  the  formula: 

cos  ju'  =    —  sin  T'  •  cos  (\j/f  —  TT)  ; 

and  it  enables  us  to  find  the  magnitude  of  the  angle  \f/'. 

220.  In  the  special  case  of  a  Plane  Refracting  Surface,  for  which 
the  centre  C  is  the  infinitely  distant  point  of  the  optical  axis,  the  plane 
surface  must  be  taken  for  the  ^z-plane,  and  hence  the  three  points 
B,  H  and  Hf  coincide.  Accordingly,  for  this  special  case  we  have: 


And  since  the  incidence-normal  is  parallel  to  the  optical  axis,  we  have 
also  a  =  r,  a!  =  T';  and,  therefore, 

.      n   . 
sin  r  =  —  sm  r 

is  the  equation  for  determining  the  magnitude  of  the  angle  T'.  More- 
over, since  both  the  incident  and  refracted  rays  lie  in  the  plane  of 
incidence,  containing  the  incidence-normal,  which  here  is  parallel  to 
the  #-axis,  the  projections  of  these  rays  on  the  ^3-plane  must  coincide 
with  each  other;  and,  therefore, 


By  means  of  the  above  equations,  we  can  find  the  four  parameters 
pr,  TT',  T'  and  \f/'  of  a  ray  refracted  at  a  Plane  Surface. 


CHAPTER    X. 

TRIGONOMETRIC   FORMULAE  FOR   CALCULATING  THE  PATH  OF  A  RAY 

REFRACTED  THROUGH  A  CENTERED  SYSTEM  OF  SPHERICAL 

REFRACTING  SURFACES. 

CASE  I.    WHEN  THE  RAY  LIES  IN  THE  PLANE  OF  A  PRINCIPAL  SECTION. 

ART.  66.     CALCULATION-SCHEME  FOR  THE  PATH  OF  A  RAY  LYING  IN  THE 

PLANE  OF  A  PRINCIPAL  SECTION  OF  A  CENTERED  SYSTEM 

OF  SPHERICAL  REFRACTING  SURFACES. 

221.  In  order  to  compute  the  path  of  a  ray,  which  undergoes  suc- 
cessive refractions  (or  reflexions)  at  a  series  of  centered  spherical 
surfaces,  whose  optical  axis  lies  in  the  plane  of  incidence  of  the  first 
surface,  it  is  sufficient  to  obtain  the  set  of  formulae  for  one  of  the 
surfaces,  say,  the  &th;  for  the  calculation  will  consist  merely  in  the 
repeated  employment  of  this  set  of  formulae  for  each  of  the  surfaces 
in  succession.  We  shall  require  also  a  so-called  "transformation- 
formula",  which  will  enable  us  to  pass  from  one  surface  to  the  next. 

The  points  where  the  ray  crosses  the  optical  axis,  before  and  after 
refraction  at  the  kth  surface,  will  be  designated  by  L'k_^  Lk>  respect- 
ively, and  the  abscissae  of  these  points,  with  respect  to  the  vertex  Ak 
(Fig.  124)  of  the  kth  surface,  will  be  denoted  by  vk,  v'k\  thus, 


The  "transformation-formula",  by  which  we  transform  from  the  origin 
of  abscissae  Ak  of  the  kth  surface  to  the  origin  Ak+l  of  the  next  sur- 
face is: 

dk  =  Vk  -Vk+ll 

where  dk  —  AkAk+l  denotes  the  so-called  "thickness"  of  the  medium 
which  lies  between  the  kth  and  (k  +  i)th  spherical  surfaces,  and 
whose  absolute  index  of  refraction  is  denoted  by  n'k. 

The  radius  of  the  kth  surface  will  be  denoted  by  rk  (  =  AkCk); 
and  the  angles  of  incidence  and  refraction  at  the  kth  surface  will  be 
denoted  by  ak,  a'k.  The  "slope  "-angles  of  the  ray  before  and  after 
refraction  at  the  kth  surface  will  be  denoted  by  0^,  6k,  respectively; 
thus, 


316 


221.]  Path  of  Ray  through  Centered  Optical  System.  317 

The  following  system  of  formulae  (see  §  211)  may  now  be  written: 

sin  ak  =  (  I J  -sin  O'k, 


sm  ak  =  — —  sin  aft, 


sn 


(215) 


In  these  formulae  we  must  give  k  in  succession  all  integral  values 
from  k  =  i  to  k  =  m,  where  m  denotes  the  total  number  of  spherical 


FIG.  124. 

PATH  OF  A  RAY  IN  A  PRINCIPAL  SECTION  OF  A  CENTERED  SYSTEM  OF  SPHERICAL  REFRACTING 
SURFACES. 

AkLk'-\  =  Vk,    AkLk'  =  Vk'  ,    AkCk  =  rk,    DkBk  =  hk,    Ak-\Ak  —  dk-\,    Bk-\Bk  =  8*-i, 
'-i  =  Ik,    BkL£,ti,     L  Ak-\Lk'-\Bk-\  =  <V-i, 


surfaces.  If  we  know  the  values  of  the  constants  nk_ly  nk  and  rk 
for  the  kth  refracting  surface,  and  if  we  have  determined  the  ray-co- 
ordinates vkJ  0^_!  of  the  ray  incident  on  this  surface,  the  first  lour 
of  the  formulae  (215)  above  enable  us  to  find  the  ray-co-ordinates 
v'k,  Q'k  of  the  ray  after  refraction  at  the  &th  surface;  whereas  the  last  of 
these  formulae  enables  us  to  pass  to  the  next  surface,  provided  we 
know  the  axial  "thickness"  dk  between  the  &th  and  the  (k  +  i)th 
surfaces.  Thus,  having  found  the  magnitude  vk+l,  we  can  proceed 
to  make  the  same  calculation  for  the  (k  +  i)th  surface,  and  so  on, 
until  we  obtain,  finally,  the  co-ordinates  of  the  emergent  ray,  viz., 
v'm  =  AmL'm,  &'m  =  /.AmL'mBm.  An  actual  numerical  example,  illus- 
trating the  calculation-process  by  means  of  formulae  (215),  is  given 
in  Art.  67. 


318  Geometrical  Optics,  Chapter  X.  [  §  224. 

222.  We  have  also  a  number  of  other  relations,  which  are  often 
very  useful  and  convenient.  Thus,  if  the  symbols  <pk,  hk,  lk,  l'k  have  the 
following  significations : 

kAk,  hk  =  DkBk,  lk  =  BjJL'^j  l'k  =  BkL'k1 


where  the  letters  designate  the  points  shown  in  the  diagram  (Fig.  124), 
we  have  immediately,  in  connection  with  formulae  (215): 

^ -*»/««»  Vi.     lt=    -hjsm0k,l  (216) 

//       _    '       /)' 

<Pk  —  ak  ~  "k-i  —  ak  "~  Ok- 

223.  If  the  position  of   the  ray  is  defined   by  its  "slope"  (0*_i) 
and  its  intercept  bk  (=  CkHk)  on  the  "central   perpendicular",  we 
obtain  (see  §  211,  Note  4)  the  following  calculation-scheme: 

bk  •  cos  0£_1 
sin  ak  =  -  

sin  a'k  =  — ^P1-  sin  ak,     6k  =  0^_x  +  a'k  —  o^,  \  (217) 

,       w^  cos  0^ 
*        ?4     cos  0£     *' 

together  with   the  following   "transformation-formula",   for  passing 
from  the  &th  to  the  (k  +  i)th  surface: 

bk+i  —  b'k  +  ak  '  tan  0^;  (218) 

where 

denotes  the  abscissa  of  the  centre  Ck+l  with  respect  to  the  centre  Q,; 
that  is,  ak  =  Ck  Ck+l. 

The  relation  between  the  intercepts  bk  and  vk  is  given  by  the  fol- 
lowing formula: 

bk  =  (rk  —  vk)  tan  0'^.  (220) 

ART.  67.     NUMERICAL  ILLUSTRATION. 

224.  By  means  of  the  formulae  (151),  we  can  find  the  position  of 
the  image-point  M*m ,  which  corresponds  by  Paraxial  Rays  with  the 
axial  object-point  Ml  (or  L^,  and  by  means  of  formulae  (215)  above 
we  can  determine  the  position  on  the  axis  of  the  point  L'm  where  the 


§  224.]  Path  of  Ray  through  Centered  Optical  System.  319 

extreme  outside  ray,  or  so-called  "edge-ray",  of  the  bundle  of  rays 
crosses  the  optical  axis  after  emerging  from  the  centered  system  of  m 
spherical  refracting  surfaces  :  and  thus  we  can  compute  the  longitudi- 
nal aberration  along  the  axis: 


In  practice  this  is  found  to  be  a  very  useful  way  of  computing  the 
magnitude  of  this  aberration,  especially  in  the  case  of  optical  systems 
of  comparatively  wide  apertures,  to  which  the  theory  of  aberrations 
of  the  first  order  does  not  apply  very  well.  By  repeated  trials  in  this 
fashion,  it  is  possible,  also,  to  discover  how  the  thicknesses  and  radii 
will  have  to  be  altered  so  that,  for  example,  the  edge-ray  will  emerge 
so  as  to  cross  the  optical  axis  at  a  point  L'm1  which  coincides,  very 
nearly  at  least,  with  the  so-called  "GAUSsian"  image-point  M'm\  in 
which  case  for  this  pair  of  rays  (that  is,  for  a  paraxial  ray  and  the  edge- 
ray),  we  shall  have  vm  —  um  =  o,  approximately.  In  the  design  of 
optical  instruments  this  calculation-process  is  found  to  be  extremely 
serviceable.  In  order  to  exhibit  the  use  of  the  formulae,  we  shall  give 
here  a  rather  simple  numerical  illustration. 

For  this  purpose,  we  shall  select  an  example  given  in  TAYLOR'S 
System  of  Applied  Optics  (London,  1906),  page  101,  as  follows: 

The  optical  system  is  a  large  Telescope  Object-Glass,  of  12-in. 
aperture  (h^  =  6  in.),  consisting  of  a  biconvex  crown-glass  lens  and 
a  biconcave  flint-glass  lens,  with  the  following  radii  and  thicknesses 
(all  measured  in  inches)  : 

'i  =  +  59-8;    </!  =  +i;    r2  =  -  90.15;    d2  =  0.013; 
r3  =  -  84.7;     dz  =  +  i  ;     and     r4  =  +  410. 

The  values  of  the  refractive  indices,  for  rays  corresponding  to  the 
FRAUNHOFER-Line  C,  are  as  follows: 

n\  =  n2  =  n\  ~  J  I     n\  —  1.5146;     n'3  =  i.  6121. 
The  incident  rays  are  parallel  to  the  optical  axis,  so  that 

u\  —  v\  ~  °°>  and    0i  =  o. 

According  to  the  first  of  formulae  (216),  we  have,  therefore,  in  such 
a  case  as  this: 

sin*!  =-L,  (vl  =  GO),  (221) 

ri 

which  is  the  formula  we  must  employ  here  in  order  to  determine  the 
value  of  of. 


320 


Geometrical  Optics,  Chapter  X. 


[  §  224. 


The  calculation  will  be  divided  into  two  parts,  as  follows: 

(1)  The  calculation  of  the  Path  of  a  Paraxial  Ray,  by  means  of 
formulae  (151);  and 

(2)  The  trigonometric  calculation  of  the  Path  of  the  Edge-Ray  by 
means  of  formulae  (215)  above,  together  also  with  formula  (221)  above. 

The  sign  +  or  —  written  after  a  logarithm  indicates  the  sign  of  the 
number  to  which  the  logarithm  belongs.  Each  vertical  column  contains 
the  calculation  for  one  surface:  accordingly,  in  the  present  example, 
where  there  are  four  refracting  surfaces,  each  table  will  contain  four 
such  columns. 

For  the  Edge-Ray:  h^  =  6  inches,  vl  =  oo  and  0l=  o:  hence,  ac- 
cording to  formula  (221)  above,  we  have: 

Igfci  =  0-778I5I3  + 
clg  rl  =  8.2232988  + 

Ig  sin  c^  =  9.0014501  + 

This  forms  the  starting  point  for  the  calculation  of  this  ray. 
The  two  parts  of  the  calculation  follow. 

I.  PARAXIAL  RAY:  2^  =  00. 
Formulae : 


n   -  n_l 


UL 

I          I 

«&+!  Uk 


-  dk/uk 


Ig  (»i 
Ig  (ni 

clgrk 

Ig  (n'k— n'k-\) 

clg  w* 


- 

lg 


(n'k—nk—i)/rkn'k 

I/Uk 

clg  wit 


lg  dk/u'k 
i—dk/u'k 

clg  u'k 

clg  (i  —dk/u'k) 

lg  I  /Uk+l 


fc-1 

*  =  2 

*=3 

k  =  4 

8.2232988+ 
9.7114698  + 
9.8197020  + 

7.7569451  + 

0.1802980  + 

8.1573196+ 

9.7926080  + 

7.6481546+ 
0.2073920  + 

7.9372431  + 

7.9499276  + 

7.8555466+ 

8.0450343  — 
9.7114698  — 

o.ooooooo 

8.0721166  — 
9.7868224  + 
9.7926080  + 

7.3872161  + 

9.7868224  — 
o.ooooooo 

7.7544706+ 

7.7565041  + 

7.6515470— 

7.1740385- 

o.ooooooo 
+  0.0056816 

+0.0086545 
+0.0057083 

+0.0089110 
—  0.0044828 

+0.0071705 

—  0.0014929 

+0.0056816 

+0.0143628 

+0.0044282 

+0.0056776 

7.7544706+ 

o.ooooooo 

8.1572385  + 

8.1139434  + 

7.6462272  + 
o.ooooooo 

7.7541648+ 

u't  =  +  176.13077  in. 

7.7544706+ 

6.2711819+   7.6462272  + 

+0.9943184 

+0.9998133  +0.9955718 

7.7544706+ 
0.0024745  + 

8.1572385+ 

0.0000811  + 

7.6462272  + 
0.0019274+ 

7-7569451  + 

8.1573196+ 

7.6481546+ 

224.]  Path  of  Ray  through  Centered  Optical  System. 

Formula  for  the  Focal  Length  e': 


321 


lg(i  -dju,)  = 

Ig  (i  -  <y%)  =  9.9999189  + 

lg  (i  ~  dju3)  =  9.9980726  + 

clg  «1  =  7.7541648  + 

clge'  =  7.7496818  — 

e'  —  —  177.9583  inches. 

II.  EDGE-RAY:  See  formulae  (215)  of  this  Chapter. 


lg  (i  —Vk/r 
Ig  sin  Q'k—  i 

lg  sin  oik 
Ign'k-i/n'k 

lg  sin  a'k 

fc-4 


CCfc 

#'* 

lg  sin  a* 
clg  sin  Q'k 

lg  (sin  al/sin  ^1) 
sin  aJt/sin  ^A 
1  —  sin  a^/sin  ^i 

,     /         sin  a!k\ 
lg  (  i  --  ^-S7  ) 
\         sin  Ok/ 


Vk 

-dk 

Vk+l 
lg  Vk+l 

clg  n+i 


fe  =  l 

fe  =  2 

£  =  3 

fc  =  4 

0.4679039  + 
8.5340672  — 

0.2585662  + 
8.9356090- 

9.6547511  + 
8.4225418  — 

9.0014501  + 
9.8197020+ 

9.0019711  — 
0.1802980  + 

9.1941752  — 
9.7926080  + 

8.0772929  — 
0.2073920  + 

8.8211521  + 

9.1822691  — 

8.9867832  — 

8.2846849  — 

o°  o'  o" 
-S°4S'3°"*3 

-i°57'36".3 
+5°45'S5//.3 

-4°56/46".3 
+8°59/48".2 

-1°30'57'.« 

+o°4i/  4;/.S 

-5°45'3o".3 
+3°47'54".o 

+3°48'i9".o 
-8°45'  S'.3 

+4°  3'  i"-9 
-S°33'S9".7 

—  o°  49'  53".  3 
-i°  6'i3".2 

-i°57'36".3 

-4°56'46'/.3 

^iVsTj? 

-i°56'  6".5 

8.8211521  + 
1.4659328- 

9.1822691  — 
1.0643910  — 

8.9867832  — 
1.5774582  — 

8.2846849  — 
1.4715561  — 

0.2870849  — 

0.2466601  + 

0.5642414+ 

9.7562410  + 

—  1.9368 

+  1.764655 

+3.666414 

+0.570481 

+2.9368 

—0.764655 

—2.666414 

+0.429519 

0.4678744  + 
1.7767012  + 

9.8834656  — 
1.9549657  — 

0.4259276  — 
1.9278834  — 

9-6330430+ 
2.6127839  + 

2.2445756  + 

1.8384313  + 

2.3538110  + 

2.2458269  + 

+  175.6206 

+68.93365 

+225.8452 

+  176.1273 

0.013 

—   I.OOOO 

+  174.6206 

+68.92065 

+224.8452 

2.2420955  + 
8.0450343  — 

1.8383494  + 
8.0721166  — 

2.3518836+ 
7.3872161  + 

0.2871298  — 

9.9104660  — 

9-7390997+ 

—  1.9370 

—0.813703 

+0.548403 

+2.9370 

+  1.813703 

+0.451597 

4  —  wi  =  —  0.0035  inches. 


22 


322  Geometrical  Optics,  Chapter  X.  [  §  225. 

CASE  II.    WHEN  THE  PATH  OF  THE  RAY  DOES  NOT  LIE  IN  THE  PLANE  OF  A  PRINCIPAL 
SECTION  OF  THE  CENTERED  SYSTEM  OF  SPHERICAL  REFRACTING  SURFACES. 

ART.  68.     TRIGONOMETRIC   FORMULAE  OF  A.  KERBER   FOR    CALCULATING 

THE  PATH   OF  AN   OBLIQUE    RAY  THROUGH  A   CENTERED 

SYSTEM   OF  SPHERICAL  REFRACTING  SURFACES. 

225.  In  the  calculation-scheme  of  A.  KERBER1  (see  §§214  and  216) 
the  parameters  of  the  ray  before  refraction  at  the  kth  surface  of  the 
system  of  spherical  refracting  surfaces  are  the  co-ordinates  (xfft  k,  yg  *) 
and  (xit  k,  Zi,  *)  of  the  points  Gk  and  Ik  where  the  ray  crosses  the  two 
meridian  co-ordinate  planes,  viz.,  the  xy-  plane  and  the  xs-plane,  re- 
spectively; and,  similarly,  the  parameters  of  the  ray  after  refraction 
at  this  surface  are  the  co-ordinates  (x9tkl  y9tk)  and  (x'ijk,  zitk)  of  the 
points  G'k  (or  Gk+l)  and  I'k  (or  Ik+l)  where  the  refracted  ray  crosses 
the  xy-  and  #z-planes,  respectively.  In  order  to  obtain  the  refract- 
ion-formulae for  the  kth  surface,  we  have  merely  to  affix  to  the  sym- 
bols in  formulae  (204)  the  ^-subscript  to  indicate  that  the  formulae  are 
to  be  applied  to  the  kth  refracting  surface. 

It  will  also  be  necessary  to  obtain  a  system  of  "  Transformation- 
Formula",  whereby,  having  ascertained  the  values  of  the  co-ordinates 
(x'ff,*,  3V,*)  and  (*i,*»  4*)  of  tne  points  Gv(or  Gk+l)  and  I'k  (or  7&+1),  re- 
ferred to  the  centre  Ck  of  the  kth  surface  as  origin,  we  can  compute 
the  values  of  the  co-ordinates  (xfftk+i,  yg,k+i)  and  (xitk+i,  zijA;+1)  of  these 
same  points  referred  to  the  centre  Ck+l  of  the  (k  +  i)th  surface  as 
origin.  This  shifting  of  the  origin  along  the  x-axis  will  affect  only 
the  #-co-ordinates.  Thus,  evidently,  we  shall  have: 


where 

%  =  CkCt+l  =  dk  +  rk+l  -  rk.  (222) 

Accordingly,  in  the  Calculation-Scheme  of  A.  KERBER,  we  have  the 
following  system  of  formulae: 

(i)  Refraction-  Formula  for  Finding  the  Values  of  the  Parameters 
xff,  kj  yg,  k,  x't,  k  and  z'it  k  of  the  Ray  After  Refraction  at  the  kth  surface  : 


tan  <p9t  k  =  —  yg,k/xffi  ^     tan  <pttk  =  —  zf)  */ #<,  & ; 
tan  €*_,  = 


~  xff,k 
> 
tan  5^  =  2f-*'cosc^- 


(223) 


1  A.  KERBER:  Beitraege  zur  Dioptrik,  Heft  II  (Leipzig,  GUSTAV  FOCK,  1896),  pages  5-8. 


§  226.] 


Path  of  Ray  through  Centered  Optical  System. 


323 


Xg,k 


cos<pgtk'  cos  pi,  k 

cos  Bgt  fc-i  =  cos  (e£_i  —  <pff,  k)  •  cos  d'k-i ; 

rk  -  vfft  k  .      , 
sin  6i  *_i  =  -       ^—  sin  O  *_! ; 


rk  -  vg,k  . 
sm  ak  =  —        —  sm 


sm  a    =  — —  sin 


•  sm  ak 


sm 


sn 


x'g,k  =  -  (rk  -  vfgt 
y'g,k  =  —  x' 


cos  <pg,k,     x'i>k  =  -  (rk  -  vi>k)  -cos  <pttk; 
tan  v?  kj     £  k  =  —  #<  %  •  tan  ^><  k. 


(223, con- 
tinued) 


(2)  Transformation-  Formula  for  Determining  the  Parameters  xg,k+i, 
yg,k+i,  Xttk+i  and  zitk+i  of  the  Ray  Before  Refraction  at  the  (k  +  \)th 
surface  : 


k-  rk+l  - 

yff,  k+i  =  y 


xlt  k+i  = 


-  rk+1  - 


(224) 


226.     The  Initial  Values. 

The  position  of  the  ray  incident  on  the  first  surface  of  the  centered 
system  of  spherical  refracting  surfaces  will  be  defined  generally  by 
giving  the  co-ordinates  of  the  object-point  Plt  whence  the  ray  ema- 
nates, and  the  co-ordinates  of  the  point  Plf  where  the  ray  crosses 
the  plane  of  the  so-called  "Entrance-  Pupil"  (see  §  257  and  §  361). 
Usually,  it  will  be  possible  to  select  as  the  plane  of  the  principal  sec- 
tion to-plane)  the  meridian  plane  of  the  optical  system  which  con- 
tains the  object-point  Plt  so  that  this  point  will,  therefore,  coincide 
with  the  point  designated  by  Gr  If  Ml  designates  the  foot  of  the 
perpendicular  let  fall  from  Pl  on  the  optical  axis,  and  if  we  put 


the  co-ordinates  of  the  point  P,,  referred  to  a  system  of  rectangular 
axes  with  origin  at  Clt  will  be: 


=  o. 


324  Geometrical  Optics,  Chapter  X.  [  §  226. 

In  every  actual  optical  instrument  the  angular  opening  of  the 
bundle  of  "effective"  rays,  which,  emanating  from  the  object-point 
Plt  traverse  the  system  of  lenses,  is  limited  in  some  way,  usually  by  a 
"stop",  consisting  of  a  plane  screen  perpendicular  to  the  optical  axis 
with  a  circular  opening  in  it,  whose  centre  (called  the  "stop-centre") 
lies  on  the  optical  axis  of  the  instrument.  Even  when  no  screen  of 
this  description  is  employed,  the  cone  of  effective  rays  will  be  deter- 
mined by  the  rim  of  one  of  the  glasses  —  in  some  instances,  also,  by 
the  iris  of  the  eye  of  the  observer.  The  "stop"  is  not  always  situated 
in  front  of  the  entire  system  of  lenses;  it  may  lie  between  one  pair 
of  them,  or  it  may  even  be  placed  beyond  them  all.  Let  us  take  the 
most  general  case  and  assume  that  the  "stop"  is  situated  between, 
say,  the  bth  and  the  (b  +  i)th  surfaces  of  the  system  of  m  spherical 
surfaces,  and  let  us  designate  the  position  of  the  stop-centre  by  M'b. 
This  point  M'b  will  be  the  image,  formed  by  Paraxial  Rays,  after 
having  traversed  the  first  b  surfaces  of  the  system,  of  a  certain  axial 
object-point  Afx;  which  latter  point  is  the  centre  of  the  so-called 
"Entrance-Pupil".  The  transversal  plane  o^  perpendicular  to  the 
optical  axis  at  M±  (which  in  any  given  optical  system  will  always  be  a 
perfectly  definite  plane)  is  the  Plane  of  the  Entrance-Pupil.  And  the 
point  where  an  object-ray,  emanating  from  the  object-point  Plt  crosses 
this  plane  will  be  designated  here  by  Plt  as  has  been  stated  above. 
Moreover,  we  shall  put  AlMl  =  ulf  and  shall  denote  the  co-ordinates 
of  Plt  referred  to  rectangular  axes  with  Cl  as  origin,  as  follows: 

«i  ~  rlt  i\lt  Jt. 

As  has  been  remarked,  the  position  of  the  object-ray  is  usually 
given  by  assigning  the  values  of  the  magnitudes  denoted  here  by  the 
symbols  «„  rjl  and  t|lf  j^.  By  drawing  a  simple  diagram,  the  reader 
will  easily  perceive  that,  if  K  designates  the  projection  of  the  point  II 
on  the  xy-plane  (CiK  =  xitit  KIi  =  zifl),  we  have  the  following  re- 
lations : 


_  KI, 

^"Mf    ^ 

whence,  since 


i  +  A^  +  dK  =  xtli  +  TI  -  «i, 

+  A^  +  dK  =  xtt  i  +  r,  -  ulf 
and 

MlAl  +  AlMl  =  ul  —  ult 


§  227.]  Path  of  Ray  through  Centered  Optical  System.  325 

we  obtain: 

rh      Xj  i  +  r\  —  Ui      zi      xi  i  +  ri  —  Ui 


Thus,  we  obtain  the  initial  values  #<fl,  zitl  as  follows: 

(225) 


In  case  the  object-point  Pl  is  infinitely  distant,  the  object-rays  will 
constitute  a  bundle  of  parallel  rays;  and,  since,  in  general,  rilt  as  well 
as  «!,  will  be  infinite,  the  value  of  xiti,  as  given  by  the  first  of  formulae 
(225),  will  be  illusory.  Under  these  circumstances,  we  shall  require 
to  know  the  direction  of  the  object-ray,  and,  since  all  the  object-rays 
proceeding  from  one  and  the  same  point  of  the  object  are  parallel,  it 
will  be  sufficient  if  we  are  given  the  slope-angle  6j  of  that  one  of  the 
bundle  of  object-rays  which  crosses  the  optical  axis  at  the  centre  Mv 
of  the  Entrance-Pupil.1  Now,  evidently, 

tanO, 


if,  therefore,  in  the  expression  for  Xtti  given  in  (225),  we  substitute 
the  value  of  the  ratio  ujiju  as  obtained  from  this  last  equation,  and 
then  put  u\  =  ^  =  oo,  we  shall  derive  the  first  of  the  two  following 
formulae  : 

Xi  i  =  «i  —  ili  •  cot 

(226) 

The  latter  formula  is  obvious  immediately  from  the  second  of  formulae 

(225). 

ART.  69.    THE  TRIGONOMETRIC  FORMULAE  OF  L.  SEIDEL  FOR  CALCULATING 

THE   PATH   OF  AN   OBLIQUE  RAY  THROUGH  A   CENTERED 

SYSTEM  OF  SPHERICAL  REFRACTING  SURFACES. 

,  227.  Employing  here  the  same  notation  as  was  used  in  §§215 
and  219,  where  the  calculation-scheme  of  L.  SEIDEL  for  the  case  of 
the  refraction  of  an  oblique  ray  at  a  single  spherical  surface  was  given, 

1  This  will  not  be  the  Chief  Ray  of  the  bundle,  unless  the  stop-centre  coincides  with 
the  centre  of  the  Entrance-Pupil  ;  or  unless,  with  respect  to  these  two  points,  the  spherical 
aberration  of  that  part  of  the  optical  system  which  precedes  the  stop-centre  has  been 
abolished. 


326  Geometrical  Optics,  Chapter  X.  [  §  227. 

we  shall  designate  the  points  where  the  ray  crosses  the  kth  transversal 
(or  yz-)  plane,  before  and  after  refraction  at  the  kth  surface,  by  HA, 
H'k,  respectively;  and  shall  denote  the  rectangular  co-ordinates  of 
these  points  by  (o,  yhik,  zhtk),  (o,  yhtk,  z'hik),  and  their  polar  co-ordinates 
by  (pk,  TTfc),  (p'k,  O»  respectively:  the  relations  of  these  two  sets  of  co- 
ordinates being  defined  as  follows : 


=  pk-simrk,-\ 
,  ,     .       ,    f  (227) 

yh,  k   =  Pk  •  COS  TTk,       Zht  k   =  pk  -  Sin  TTjfc,  J 

where 

(228) 


(or,  in  case  the  kth  surface  is  a  reflecting  surface,  ir'k  =  irk  -f-  180°). 

The  directions  of  the  ray,  before  and  after  refraction  at  the  kth 
surface,  are  denned  by  two  pairs  of  angular  magnitudes  denoted  by 
rfc,  \f/k  and  rk,  jf£,  respectively.  Since  the  direction  of  the  ray  after 
refraction  at  the  kth  surface  is  identical  with  its  direction  before  re- 
fraction at  the  (k  +  i)th  surface,  we  have: 

r'k  =  T*+I>     *I  =  ifcH-iJ  (229) 

which  are,  therefore,  the  "Transformation-Formulae"  for  SEIDEL'S 
Direction-Parameters.  These  Direction-Parameters  are  defined,  pre- 
cisely as  in  §  215,  by  the  following  formulae: 


i,  k+l 


, 

tan  TJ.  =  —  --   —  77  =  tan  Tk,l  = 


(230) 


It  remains  to  obtain  L.  SEIDEL'S  Formulae  for  the  transformation 
from  the  parameters  irk,  p'k  to  the  parameters  TTA+I,  pk+1;  which  we 
proceed  to  do. 

If  the  Direction-Cosines  of  the  ray  after  refraction  at  the  &th  surface 
are  denoted  by  a,  /3,  7,  then,  precisely  as  in  §  215,  we  have: 

|8  7 

-  =  —  tan  rk  •  cos  \l/k,       -  —  —  tan  rk  •  sin  \frk ; 

and  since  this  ray  goes  through  the  two  points  Hk  and  Hk+l,  whose 
rectangular  co-ordinates,  referred  to  the  centre  of  the  &th  surface  as 
origin  are: 

(o,  p'k  •  cos  irk,  pk  •  sin  TTA.)     and     (ak,  pk+l  -  cos  irk+l,  pk+l  •  sin  irk+l) , 


§  227.]  Path  of  Ray  through  Centered  Optical  System, 

respectively,  we  have: 


327 


irk+l  -  p'k-sm 


a   '  0  7 

Eliminating  a,  |8,  7  from  these  two  sets  of  equations,  we  obtain: 

pk+l  •  cos  irk+l  -  p'h.  •  cos  irk  =  -  ak  -  tan  rk  •  cos  ^, 
pk+l-sin  irk+l  —  /Vsin  TT^.  =  —  a*  -tan  r^-sin  ^. 

Combining  these  equations,   we  obtain  easily  the  Transformation- 
Formulae  of  L.  SEIDEL/  as  follows: 


p 


sm        - 


'*  -  **+i)  =  Pk-cos  ($'k  -  O  -  «* 


(231) 


Accordingly,  in  the  Calculation-Scheme  of  L.  SEIDEL  for  the  re- 
fraction of  an  oblique  ray  through  a  centered  system  of  spherical 
surfaces  we  have  the  following  formulae  (see  formulae  (213)  and  (214)): 

(i)  Determination  of  the  Position  of  the  Point  H'k  by  means  of  its 
Polar  Co-ordinates  (pk,  O  : 


cos  Mi  =  —  sm  rfc_j  •  cos 
sinat  =  £fc*8tnnfc/fv; 


sin  dj,  = 


Pk  = 


sin  offc ; 


sin  a, 


Sm 


(232) 


sm  ak  nk    sin  M& 

(2)  Determination  of  the  Direction  (r'k,  ^D   of  the  Refracted  Ray: 

,          ,  ,-         .        sin  M!  ,  , , '  N 

sin  (&,  —  TT J  =  — sin  r^.j  •  sin  (y^|  —  TT^)  ; 


sin 


tan  (irk  - 


sin     _ 

sin  r^  •  sin  (\l/'k  —  Q 


tan  rA  =  tan  ,T^._I  • 


sn 


sn         - 


(233) 


1  L.  SEIDEL:  Trigonometrische  Formeln  fur  den  allgemeinsten  Fall  der  Brechung  des 
Lichtes  an  centrierten  sphaerischen  Flaechen  :  Sitzungsber.  der  math.-phys.  Cl.  der  kgl. 
bayr.  Akad.  der  Wissenschaften,  vom  10.  Nov.  1866.  Reprinted  in  Beilage  III  of  STEIN- 
HEIL  &  VOIT'S  Handbuch  der  angewandten  Optik,  Bd.  I  (Leipzig,  B.  G.  TEUBNER,  1891), 
pages  257-270. 


pk+l-sm  (t'k  -  irk+1)  =  pk-sin  (f'k  -  TT J ; 
pk+l-cos  ($'k  -  71-*+!)  =  p'k-cos  ($'k  -  irk)  - 


328  Geometrical  Optics,  Chapter  X.  [  §  228. 

(3)   Transformation- Formula  for  finding  the   parameters  pk+l,  irk+l 
of  the  Ray  Before  Refraction  at  the  (k  +  i)th  Surface: 


>34) 


228.     Seidel's  "Control"  Formulas. 

In  order  to  check  the  numerical  work  from  time  to  time,  and  thereby 
to  avoid  the  disagreeable  necessity,  in  case  of  arithmetical  errors,  of 
having  to  repeat  sometimes  a  very  considerable  portion  of  the  calcu- 
lation, L.  SEIDEL  has  proposed,  in  connection  with  the  above  formulae, 
several  so-called  "Control"  Formula,  the  first  of  which  is  as  follows: 

sin  nh  -  sin  /4         sin  nk  •  sin  r^         sin  (ak  —  ak) 

' .  — ^z r ^^ (2^  C  i 

sin  Wft_j  —  TTA)        sin  (\f/k  —  7rk)        sin  (^»_|  —  ^J* 

The  equality  of  the  two  expressions  on  the  left  follows  from  the  first 
of  formulae  (232);  and  the  equality  between  each  of  these  and  the 
third  expression  can  be  deduced  easily  from  the  first  of  formulae  (232) 
and  the  first  of  formulae  (233).  Accordingly,  this  "control"  formula 
(235)  serves  to  test  only  the  accuracy  of  computations  by  these  for- 
mulae from  which  it  is  derived. 

The  values  of  the  sines  of  the  angles  of  incidence  and  refraction 
are  checked,  along  with  the  value  of  p't  by  the  double  calculation  of 
this  latter  magnitude  by  means  of  the  two  expressions  for  pf  in  formulae 
(232).  But  as  it  is  possible  that,  even  though  we  have  found  the 
correct  value  of  the  sine  of  an  angle,  an  error  may  be  introduced  in 
determining  the  value  of  the  corresponding  angle  itself,  or  that  a 
mistake  may  be  made  in  obtaining  the  difference  a  —  a.',  thereby 
involving  also  a  mistake  in  the  value  obtained  for  the  angle  p! ,  and  as 
the  "control"  formula  (235)  would  not  enable  us  to  detect  an  error  of 
any  of  these  kinds,  SEIDEL  suggests  also  a  second  "control"  formula, 
as  follows: 

-  .      .  =  — .      —     — . 

sin  ak  •  sin  ak  Wk—i        ^k 

which  is  a  simple  consequence  of  the  Law  of  Refraction.  The  magni- 
tude on  the  right  is  constant  for  all  rays  of  the  same  wave-length 
refracted  between  the  same  two  media;  so  that  in  case  the  calculation 
has  to  be  made  for  a  number  of  such  rays  (as  usually  happens  in  such 
calculations),  it  will  not  be  necessary  to  calculate  at  all  the  value  of 


§  229.]  Path  of  Ray  through  Centered  Optical  System.  329 

the  left-hand  side  of  the  equation,  but  it  will  be  sufficient  merely  to 
see  that  the  values  of  the  expressions  on  the  right  are  the  same  for  all 
the  rays.  Moreover,  in  the  usual  case  of  an  optical  system  consisting 
of  a  series  of  glass  lenses,  each  surrounded  by  air,  where,  therefore, 
the  ray  proceeding  from  a  medium  (n)  into  a  medium  (nf),  emerges 
again  into  the  medium  (n),  the  values  of  the  constant  on  the  right- 
hand  side  of  (236)  for  two  successive  refracting  surfaces  will  be  equal 
in  magnitude,  but  opposite  in  sign ;  and  in  such  a  case  it  will  merely 
be  necessary  to  calculate  the  values  of  the  expression  on  the  left- 
hand  side  for  each  surface,  and  see  that  the  condition  above-mentioned 
is  fulfilled. 

Finally,  a  third  "control  "  formula,  deduced  from  the  two  trans- 
formation-formulae (231),  is  as  follows: 


sin  (^  -  irk)       sin  (irk  -  7rk+l)       sin  (^  -  7rk+l)  ' 


(237) 


In  STEINHEIL  &  VOIT'S  Handbuch  der  angewandten  Optik,  I.  Bd. 
(Leipzig,  B.  G.  TEUBNER,  1891),  the  reader  will  find  numerous  complete 
calculations  by  means  of  the  trigonometric  formulae  of  L.  SEIDEL. 

229.     The  Initial  Values. 

The  position  of  the  object-ray  will  usually  be  defined  by  the  posi- 
tion of  the  object-point  P^u^  —rlt  y^  o)  and  the  position  of  the  point 
Pl(ul  —  rlt  T^,  5i)  where  the  ray  crosses  the  plane  of  the  Entrance- 
Pupil  (see  §  226).  This  ray  crosses  the  first  transversal  (or  yz-)  plane 
at  #1(0,  yhl,  zkti)  and  the  horizontal  #2-plane  at  /i  (#<,,,  o,  z^i). 
The  positions  on  the  #-axis  of  the  points  designated  below  by  C,,  Mlt 
Afj  and  K  are  defined  as  follows: 


By  drawing  a  figure,   the  following  relations  will  be  immediately 
obvious  : 


Here 

M,K  =  AM,  +  Aid  +  dK  =  *i,i  +  *  -  «i, 

MlCl  =  MlAl  +  AlCl  =  rx  -  ult 
and 

MlMl  =  M^A\  +  AlMl  =  UT  —  ttt; 

and  if  for  xit  i  we  substitute  its  value  as  given  by  the  first  of  formulae 


330  Geometrical  Optics,  Chapter  X.  [  §  229. 


(225),  we  obtain: 

T1l(u1  -  rt)  -  ihfo  -  r,) 


(238) 


By  means  of  formulae  (238),  together  with  (225),  we  can  determine 
now  the  magnitudes  of  the  direction-parameters  (rlt  i/^)  of  the  object- 
ray;  for  according  to  the  definition-formulae  of  these  angles  we  have: 


.  +  (*..-*..) 

yh,  i  %i,  i 

and,  consequently: 


The  initial  values  plt  TI  of  the  other  two  SEiDEL-parameters  may  be 
determined  by  the  equations  : 


1  (240) 

fill 


pl  •  sin  (ft  —  TTT)  =  =*=  17 1  •  sin  ft, 

/>!  •  cos  (ft  —  TTj)  =  =*=  r/L  •  cos  ft  —  (rx  —  wj  •  tan 
wherein  the  upper  sign   must  be  used  in  case  the  object-point  lies 
above  the  optical  axis,  and  the  lower  sign  in  the  opposite  case. 

In  the  special  case  when  the  object-point  PI  is  the  infinitely  distant 
point  of  the  object-ray,  then,  in  general,  both  rji  and  HI  will  be  infinite. 
In  this  case,  instead  of  being  given  the  co-ordinates  ult  r^,  we  shall  be 
given  the  direction  of  the  ray — which  will  usually  be  done  by  assigning 
the  value  of  the  slope-angle  81  of  that  one  of  the  bundle  of  parallel 
object-rays  which  crosses  the  optical  axis  at  the  centre  Ml  of  the 
Entrance-Pupil,  and  which,  therefore,  crosses  the  first  central  transver- 
sal plane  at  a  point  whose  distance  from  the  optical  axis  is : 

If  iji,  jji  denote  the  co-ordinates  of  the  point  where  the  general  object- 
ray  lying  outside  the  plane  of  the  principal  section  crosses  the  plane 
of  the  Entrance-Pupil,  we  shall  have  in  this  case  the  following  formulae 
for  determining  the  parameters  pi,  irii 

y?i, i  —  pi' cos TI  =  t|i  -j-  (TI  —  Ui)  •  tan  81, 
ZH, i  —  pi' sin  TI  =  Ji 

Evidently,  also,  for  the  case  of  an  infinitely  distant  object-point,  we 
have  ft  =  o°  or  180°  and  n  =  =*=  81. 


CHAPTER    XI. 

GENERAL  CASE  OF  THE  REFRACTION  OF  AN  INFINITELY  NARROW 

BUNDLE  OF  RAYS  THROUGH  AN  OPTICAL  SYSTEM. 

ASTIGMATISM. 

ART.  70.     GENERAL  CHARACTERISTICS   OF  A   NARROW   BUNDLE   OF  RAYS 
REFRACTED  AT  A  SPHERICAL  SURFACE. 

230.     Meridian  and  Sagittal  Rays. 

To  an  infinitely  narrow  homocentric  bundle  of  incident  rays  re- 
fracted (or  reflected)  at  a  spherical  surface  there  corresponds,  in 
general,  an  astigmatic  bundle  of  refracted  (or  reflected)  rays,  which, 
provided  we  neglect  magnitudes  of  the  second  order  of  smallnesSj  is 
characterized  by  the  following  properties: 

The  chief  ray  u'  of  the  bundle  of  refracted  rays  is  that  one  of  the 
refracted  rays  which  corresponds  to  the  chief  ray  u  of  the  bundle  of 
incident  rays.     All  the  refracted  rays  meet  two  infinitely  short  straight 
lines,  the  so-called  Image- Lines  (§  47),  which  lie  in  two  perpendicular 
planes  both  containing  the  refracted  chief  ray  u' ,  and  which  are  perpen- 
dicular to  u'.    These  two  planes  are  the  planes  of  Principal  Curvature 
of  the  element  of  the  refracted  wave-surface  at  any  point  P'  of  the  re- 
fracted chief  ray  u' ,  which  pierces  the  surface-element  at  P'  normally, 
and  their  traces  on  the  element  of  wave-surface  at  P'  are  two  elements 
of  arc  intersecting  at  right  angles  at  P' .     The  two  pencils  of  rays  of 
the  bundle  of  refracted  rays  which  lie  in  the  planes  of  Principal  Curva- 
ture have  their  vertices  on  the  refracted  chief  ray  u'  at  the  centres 
of  curvature  S'  and  S'.     Thus,  to  an  object-point  S  lying  on  the 
incident  chief  ray  u,  which  is  the  vertex  of  an  infinitely  narrow  homo- 
centric  bundle  of  incident  rays,  correspond  two  image-points  S',  S' 
lying  on  the  refracted  chief  ray  u' ,  which  we  shall  call  the  Primary 
and  Secondary  Image-Points,  respectively.     The  two  Image-Lines  are 
perpendicular  to  the  refracted  chief  ray  u'  at  these   Image-Points. 
Thus,  the  I.  Image-Line  is  perpendicular  to  the  refracted  chief  ray 
at  S' ,  and  lies  in  the  plane  of  Principal  Curvature  of  the  refracted 
wave-surface  for  which  the  II.  Image-Point  S'  is  the  centre  of  curva- 
ture; and,  similarly,  the  II.  Image-Line  is  perpendicular  at  S'  to  the 
chief  refracted  ray  u',  and  lies  in  the  plane  of  Principal  Curvature  of 
the  refracted  wave-surface  for  which  the  I.  Image-Point  S'  is  the  centre 
of  curvature. 

331 


332  Geometrical  Optics,  Chapter  XI.  [  §  230. 

When  the  plane  determined  by  the  chief  rays  u,  u',  which  we  shall 
call  the  Plane  of  Incidence,  is  at  the  same  time  a  plane  of  Principal 
Curvature  of  the  refracted  wave-surface,  one  of  the  image-lines  will 
lie  in  this  plane,  and  the  other  will  lie  in  a  plane  perpendicular  to  the 
plane  of  incidence. 

The  special  problem  which  we  have  to  consider  presents  a  compara- 
tively simple  case;  for,  since  the  refracting  surface  is  spherical,  the 
two  systems  of  incident  and  refracted  rays  are  symmetrical  about  an 
axis.  Thus,  if  C  designates  the  centre  of  the  spherical  refracting 
surface,  not  only  this  surface  but  the  incident  and  refracted  wave- 
surfaces  as  well  are  surfaces  of  revolution  around  the  straight  line 
5  C  as  axis.  The  plane  of  incidence  u  C,  containing  the  common  axis 
of  these  three  surfaces  of  revolution,  is  a  meridian  plane  of  each  one 
of  these  surfaces,  and  is,  therefore,  also  a  plane  of  Principal  Curvature; 
so  that  one  of  the  Image-Lines  will  lie  in  the  plane  of  incidence, 
and  the  other  will  lie  in  the  plane  perpendicular  to  the  plane  of  inci- 
dence which  contains  the  refracted  chief  ray  u' .  According  to  the 
usage  of  most  writers  on  Optics,  we  shall  designate  the  latter  as  the 
I.  Image-Line  and  the  former  as  the  II.  Image-Line.1  The  II.  Image- 
Line  is  perpendicular  to  the  chief  refracted  ray  u'  at  the  point  Sf 
where  this  ray  crosses  the  axis  of  symmetry  SC. 

Thus,  in  the  case  of  an  infinitely  narrow  homocentric  bundle  of 
incident  rays  refracted  at  a  spherical  surface,  the  directions  of  the 
Image-Lines  of  the  astigmatic  bundle  of  refracted  rays  will  depend 
only  on  the  position  and  direction  of  the  chief  refracted  ray  u' \  so 
that  to  a  range  of  object-points  lying  on  a  given  incident  chief  ray  u 
there  will  correspond  a  series  of  parallel  I.  Image-Lines  .and  a  series 
of  parallel  II.  Image-Lines. 

The  planes  of  Principal  Curvature  of  the  wave-surface  determine 
two  principal  sections  of  the  infinitely  narrow  bundle  of  rays.  The 
plane  of  incidence  u  C,  which  in  the  case  of  a  spherical  refracting  sur- 
face coincides  with  one  of  these  planes,  cuts  the  infinitely  narrow 
homocentric  bundle  of  incident  rays  and  the  corresponding  astigmatic 
bundle  of  refracted  rays  in  a  pencil  of  incident  rays  with  its  vertex 
at  the  Object-Point  5  and  in  a  pencil  of  refracted  rays  with  its  vertex 
at  the  I.  Image-Point  S' '.  These  are  the  so-called  Meridian  Rays; 
since  the  plane  of  incidence  uC  is  at  the  same  time  a  meridian  plane 
of  the  spherical  refracting  surface. 

If  the  incident  chief  ray  u  is  supposed  to  be  revolved  about  SC 

1  Some  writers,  however,  for  example,  LIPPICH,  use  the  contrary  method  of  designating 
these  lines. 


§231.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  333 

as  axis  through  an  infinitely  small  angle  to  one  side  and  the  other  of 
its  actual  position,  it  will  coincide  in  succession  with  all  the  rays 
which  lie  on  the  surface  of  a  right  circular  cone  of  which  SC  is  the 
axis  and  the  straight  line  SB  (where  B  designates  the  point  where 
the  chief  ray  meets  the  refracting  surface)  is  an  element.  The  corre- 
sponding refracted  rays  will  likewise  lie  on  the  surface  of  a  right  cir- 
cular cone  generated  by  the  revolution  of  the  chief  refracted  ray  BSf 
about  the  same  line  as  axis.  Provided  we  neglect  infinitely  small 
magnitudes  of  the  second  order,  this  group  of  incident  rays  may  be 
regarded  as  lying  in  a  plane  TT  which  contains  the  incident  chief  ray 
and  is  perpendicular  to  the  plane  of  incidence  uC  (or  TT);  and,  simi- 
larly, the  corresponding  refracted  rays  may  also  be  regarded  as  lying 
in  a  plane  TT'  which  contains  the  chief  refracted  ray  u'  and  is  likewise 
perpendicular  to  the  plane  uC.  These  planes  are  evidently  tangent 
to  the  conical  surfaces  generated  by  the  revolution  of  u,  u'  around  SC 
as  axis.  Following  the  usage  of  most  modern  writers,  we  shall  call 
the  incident  and  refracted  rays  lying  in  the  planes  TT,  TT',  respectively, 
the  Sagittal  Rays.1 

231.  Different  Degrees  of  Convergence  of  the  Meridian  and  Sagit- 
tal Rays. 

The  diagram  (Fig.  125)  shows  a  meridian  section  of  the  spherical 
refracting  surface  ju  containing  the  chief  incident  ray  u  and  the  chief 


FIG.  125. 

CONVERGENCE  OF  MERIDIAN  RAYS  AFTER  REFRACTION  AT  A  SPHERICAL  SURFACE.    All  the 
lines  in  the  figure  lie  in  the  plane  of  a  meridian  section  of  the  refracting  sphere. 

refracted  ray  u',  the  plane  of  the  diagram  being,  therefore,  the  Plane 
of  Incidence.  The  numerals  i  and  2  in  the  figure  are  used  to  desig- 
nate two  points  of  the  meridian  section  of  the  spherical  surface  both 
very  close  to  the  incidence-point  B  of  the  chief  incident  ray  and  lying 

1  "  Sagittal "  is  a  term  borrowed  from  Anatomy.  Many  writers  use  the  antonym 
"  tangential  "  instead  of  "  meridian  ".  On  the  other  hand,  some  writers,  who  use  the 
term  "  meridian  ",  prefer  to  be  more  consistent  and  use  therefore  the  word  "  equatorial " 
instead  of  "  sagittal  ". 


334  Geometrical  Optics,  Chapter  XI.  [  §  232. 

on  opposite  sides  of  this  point.  Thus,  Si,  S2  belong  to  the  pencil  of 
meridian  incident  rays.  After  refraction,  these  rays  will  intersect  the 
chief  refracted  ray  u'  in  the  points  designated  in  the  figure  by  S" ,  S'" , 
which,  while  they  are  infinitely  close  together,  cannot,  in  general,  be 
regarded  as  coincident  unless  we  neglect  infinitesimals  of  the  first 
order.  In  fact,  the  position  of  the  I.  Image-Point  S'  depends  on  the 
arc  Bi,  so  that  for  different  rays  of  the  meridian  pencil  we  shall  obtain 
values  of  BS'  which  differ  from  each  other  by  magnitudes  of  the  same 
order  of  smallness  as  the  arc  Bi.  Hence,  the  convergence  of  the  re- 
fracted rays  in  the  meridian  section  is  said  to  be  a  "convergence  of  the 
first  order". 

The  convergence  of  the  refracted  rays  in  the  sagittal  section  is  of 
a  higher  order  than  the  first.     Thus,  in  Fig.  126,  which  is  the  corre- 


FIG.  126. 

CONVERGENCE  OF  SAGITTAL  RAYS  AFTER  REFRACTION  AT  A  SPHERICAL  SURFACE.  The  plane 
of  the  paper  represents  a  meridian  section  of  the  refracting  sphere.  The  points  designated  in  the 
diagram  by  the  letters  S.  Sf,  B  and  C  lie  in  this  plane.  The  points  designated  by  the  Roman 
Numerals  I  and  II  are  both  infinitely  near  to  the  point  B;  these  points  lie  on  the  line  of  inter- 
section of  the  two  planes  which  are  perpendicular  to  the  plane  of  the  paper  and  which  contain 
the  incident  ray  SB  (or  u)  and  the  corresponding  refracted  ray  BS'  (or  u'),  respectively. 

spending  diagram  for  the  case  of  the  sagittal  rays,  if  the  triangle 
SBS'  is  supposed  to  be  revolved  about  the  central  line  SCS'  as  axis 
through  an  infinitely  small  angle  above  and  below  the  plane  of  the 
paper,  the  chief  incident  ray  and  the  chief  refracted  ray  will  coincide 
in  succession  with  the  rays  of  the  bundles  of  incident  and  refracted 
rays,  respectively,  which  lie  on  the  conical  surfaces  generated  by  this 
revolution.  These  refracted  rays  all  intersect  exactly  at  the  point  S' 
and  these  rays  are  very  nearly  identical  with  the  sagittal  rays  them- 
selves. In  fact,  it  is  easy  to  see  that  the  convergence  of  the  sagittal 
rays  at  S'  is  a  "convergence  of  the  second  order",  and  is  optically 
more  effective  than  that  of  the  meridian  rays. 

232.     The  Image-Lines. 

The  astigmatic  bundle  of  rays  may  Le  regarded  as  composed  either 
of  pencils  of  meridian  rays  whose  chief  rays  all  meet  in  the  II.  Image- 
Point  S',  or  as  pencils  of  sagittal  rays  whose  chief  rays  all  meet  in  the 


§  232.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  335 

I.   Image-Point  Sf.     The  vertices  of  the  meridian  pencils  form  the 

I.  Image-Line,  and  the  vertices  of  the  sagittal  pencils  form  the  II. 
Image-Line. 

An  incident  ray  proceeding  from  S,  lying  in  the  plane  of  the  meridian 
section,  and  meeting  the  spherical  refracting  surface  at  a  point  i  a 
little  above  B  (Fig.  125)  will  be  refracted  so  as  to  intersect  the  central 
line  SC  at  a  point  slightly  to  the  left  of  3'  (Fig.  126);  and  this 
point  will  be  the  point  of  convergence  of  all  the  refracted  rays  which 
correspond  to  incident  rays  lying  on  the  conical  surface  generated  by 
the  revolution  about  SC  as  axis  of  the  incident  ray  Si.  And,  simi- 
larly, if  the  point  of  incidence  lies  in  the  meridian  section  at  a  point 
2  slightly  below  B,  the  rays  incident  on  the  spherical  surface  at  points 
in  the  arc  of  the  circle  described  by  2  when  the  figure  is  revolved  about 
5  C  as  axis  will  be  refracted  so  as  to  cross  the  central  line  at  a  point 
a  little  to  the  right  of  Sf.  Thus,  all  the  rays  of  the  infinitely  narrow 
astigmatic  bundle  of  refracted  rays  will  cross  the  central  line  S  C  within 
an  infinitely  short  piece  of  it  lying  on  either  side  of  the  II.  Image- 
Point  S'.  This  line-element  may  be  regarded,  and,  indeed,  from  a 
purely  geometrical  point  of  view,  should  be  regarded,  as  in  reality 
the  II.  Image-Line.1  However,  this  line  is  not  perpendicular  to  the 
chief  refracted  ray  uf,  and  it  is  more  convenient  and  quite  permissible 
to  consider  both  of  the  Image-Lines,  according  to  STURM'S  definition, 
as  perpendicular  to  the  chief  ray  of  the  astigmatic  bundle  (see  §  49). 
In  fact,  as  CzAPSKi2  and  others  have  pointed  out,  a  section  of  the 
bundle  of  rays  made  by  a  plane  through  S'  perpendicular  to  the  chief 
ray  u'  differs  very  little  from  a  straight  line;  the  actual  shape  of  the 
section  is  a  curve  with  two  loops,  not  unlike  a  slender  figure  8.  It  is 
easy  to  see  that  this  is  so ;  for  whereas  the  rays  of  the  sagittal  section 
proper  all  intersect  in  S',  the  rays  of  the  other  so-called  sagittal  sec- 
tions intersect  in  points  which  lie  on  the  axis  to  one  side  and  the  other 
of  5',  and,  hence,  the  rays  of  each  of  these  latter  pencils  will  meet  the 
plane,  which  is  drawn  perpendicular  to  u'  at  S',  either  before  or  after 
they  meet  each  other  at  the  vertex  of  the  pencil  on  the  central  line 
SC,  according  as  this  vertex  lies  to  the  one  side  or  the  other  of  the 

II.  Image-Point  Sf.     Thus,  we  see  that  the  section  of  the  bundle 
made  by  this  plane  opens  out  on  each  side  of  5'.     Moreover,  it  can 
very  easily  be  shown  that  the  width  of  this  section  is  a  magnitude  of 

1  See,  particularly,  L.  MATTHIESSEN:  Ueber  die  Form  der  unendlich  duennen  astig- 
matischen  Strahlenbuendel  und  ueber  die  KuMMER'schen  Modeller  Sitzungber.  der  math.- 
phys.  CL  der  koenigl.  bayer.  Akad.  der  Wissenschaften  zu  Muenchen,  xiii.  (1883),  83. 

2  S.   CZAPSKI:  Zur  Frage  nach  der  Richtung  der  Brennlinien  in  unendlich  duennen 
optischen  Buescheln:  WIED.  Ann.,  xliii.  (1891),  332-337. 


336  Geometrical  Optics,  Chapter  XI.  [  §  233. 

the  second  order  of  smallness,  and  hence  the  section  itself  may  be 
regarded  as  a  straight  line,  since  we  are  neglecting  infinitesimals  of 
the  second  order.  As  CZAPSKI  says,  the  two  8-shaped  sections,  which 
we  have  at  both  the  I.  Image-Point  5'  and  the  II.  Image-Point  iS', 
with  the  axes  of  the  8's  at  right  angles  to  each  other,  are  as  near  an 
approach  to  what  may  be  called  the  Image-Lines  of  the  astigmatic 
bundle  of  rays  as  any  other  pair  of  lines. 

ART.  71.     THE  MERIDIAN  RAYS. 

233.   Relation  between  the  Object-Point  5  and  the  I.  Image-Point  5'. 

Let  the  chief  ray  u  of  an  infinitely  narrow  homocentric  bundle 
of  incident  rays  proceeding  from  an  Object-Point  5  meet  the  spherical 
refracting  surface  /z  in  the  point  B  (Fig.  127),  and  let  the  refracted 
chief  ray  u'  corresponding  to  u  be  constructed  as  in  YOUNG'S  Con- 
struction (§  206)  by  means  of  the  concentric  spherical  surfaces  r,  rf 
described  around  C  as  centre  with  radii  equal  to  n'r/n,  nr/n',  respect- 
ively, where  C  designates  the  centre  of  the  spherical  refracting  surface, 
and  r  denotes  its  radius,  and  n,  n'  denote  the  absolute  indices  of  refrac- 
tion of  the  first  and  second  medium,  respectively.  Let  G  designate  a 
point  of  the  spherical  refracting  surface  in  the  plane  of  incidence  u  C 
and  infinitely  near  to  B,  so  that  SG  will  represent  a  secondary  ray  of 
the  pencil  of  incident  meridian  rays.  This  ray  will  meet  the  auxiliary 
spherical  surface  r  in  a  point  N  infinitely  near  to  the  point  Z  where 
the  chief  incident  ray  u  meets  this  surface,  and  the  refracted  ray  cor- 
responding to  the  incident  ray  SG  will  meet  the  spherical  surface  r' 
in  a  point  N'  infinitely  near  to  the  point  Z'  where  the  chief  refracted 
ray  u'  meets  this  surface.  The  point  of  intersection  of  this  refracted 
ray  with  the  chief  refracted  ray  will  determine  the  I.  Image-Point  S', 
wHich  is  the  vertex  of  the  pencil  of  meridian  refracted  rays. 

The  relation  between  the  Object-Point  5  and  its  I.  Image-Point  S' 
may  be  found  in  various  ways,  either  analytically  or  geometrically. 
A  very  elegant  geometrical  method,  involving  however  certain  kine- 
matical  notions  which  appear  to  be  a  little  foreign  in  a  treatise  on 
Optics,  is  given  by  L.  BURMESTER  in  his  interesting  paper,  "Homo- 
centrische  Brechung  des  Lichtes  durch  die  Linse"  (Zs.  f.  Math.  u. 
Phys.,  xl.,  1895,  321).  The  method  which  is  given  below  is  in  some 
ways  very  similar  to  that  used  by  F.  KESSLER  in  a  paper  entitled 
"  Beitraege  zur  graphischen  Dioptrik"  (Zs.  f.  Math.  u.  Phys.,  xxix., 
1884,  65-74). 

On  BC  (Fig.  127)  as  diameter,  describe  a  semi-circle  meeting  the 
chief  incident  ray  u  in  a  point  Y  and  the  chief  refracted  ray  u'  in  a 


§  233.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  337 


point  Y1 ',  and  let  us  imagine  that  straight  lines  are  drawn  connecting 
the  points  Y,  Y'  with  each  of  the  points  G,  Z  and  Z'.     Since  the 


a  ~  -H  a  I 

£°|S3 

1 14 II 
51T*| 


angles  are  infinitely  small,  we  can  write  the  following  proportions: 
ZGT5      BS         Z.GTB      BS' 


LGSB 
Z.GSB 


YZ 


/.GS'B 
ZGS'B 


BY" 
Y'Z' 


Z.NYZ      SZ'     /.N'Y'Z'       S'Z'' 
Therefore,  multiplying  each  of  the  two  upper  equations  by  the  one 

23 


338  Geometrical  Optics,  Chapter  XL  [  §  233. 

directly  below  it,  we  obtain: 

ZGYB      BS-YZ        AGY'B       BS'-Y'Z' 


ANYZ      BY-SZ'     ZTV'F'Z'      BY'-S'Z'' 

Since  we  are  neglecting  here  infinitesimals  of  the  second  order,  we  can 
regard  the  point  G  as  lying  on  the  circumference  of  the  circle  BYY'C, 
and  therefore  we  can  write: 

LGYB  =  £GY'B. 
Moreover,  since 

ABYC  =  LBY'C  =  90°, 

the  semi-circles  described  on  CZ  and  CZf  as  diameters  will  go  through 
F  and  F',  respectively.  These  semi-circles  may  also  be  regarded  as 
going  through  the  points  N  and  Nf  which  are  infinitely  near  to  Z 
and  Z',  respectively.  Accordingly, 

ANYZ  =  LNCZ  =  AN'Y'Z'; 
and,  thus,  we  obtain  the  following  relation: 

BS-YZ      BS'-Y'Z' 
BY-SZ~~  BY'-S'Z" 
or 

(BZSY)  =  (BZ'S'Y'). 

In  this  equation  the  points  designated  by  the  letters  B,  Z,  Z',  Y  and 
Yf  are  all  fixed  points  lying  on  the  given  incident  chief  ray  u  or  on  the 
corresponding  refracted  chief  ray  u' \  whereas  S'  is  the  I.  Image-Point 
on  u'  corresponding  to  an  Object-Point  5  lying  on  u.  Interpreting 
the  equation,  we  can  say: 

To  a  range  of  Object- Points  P,  Q,  R,  S,  •  •  -  lying  on  the  chief  inci- 
dent ray  u,  there  corresponds  a  protective  range  of  I.  Image- Points  P', 
Q',  R',  S',  •  •  •  lying  on  the  chief  refracted  ray  u'.  And,  moreover, 
since  the  two  ranges  have  the  incidence-point  B  in  common,  they  are  also 
in  perspective. 

That  the  points  F,  Y'  are  in  the  relation  to  each  other  of  Object- 
Point  and  I.  Image-Point  is  evident  not  only  from  the  above  equation, 
but  geometrically  also;  for  if  we  imagine  an  infinitely  narrow  pencil 
of  meridian  incident  rays  with  its  vertex  at  F,  these  rays  will  meet 
the  spherical  refracting  surface  at  points  infinitely  near  to  B  which 
may  all  be  regarded  as  lying  on  the  circumference  of  the  semi-circle 
BYC;  so  that  for  all  rays  converging  to  F,  the  angles  of  incidence, 
being  all  subtended  by  the  arc  CY,  will  all  be  equal,  and  hence,  the 


§  234.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  339 

angles  of  refraction  also  must  all  be  equal,  and  be  angles  in  the  circum- 
ference standing  on  the  arc  CYf. 

234.  The  Centre  of  Perspective  K  is  determined  by  the  inter- 
section of  the  straight  lines  YY',  ZZ'.  The  existence  of  this  point 
seems  to  have  been  recognized  first  by  THOMAS  YOUNG. l  The  point 
K  was  afterwards  found  again,  independently,  by  CoRNU2  in  1863 
and  by  LiPPiCH3  in  1878. 

Since  Z  CBZ'  =  Z  CYY',  both  being  inscribed  angles  standing 
on  the  same  arc  CYf,  it  follows  that  YY'  is  perpendicular  to  CZZ' 
at  K.  Thus,  we  have  the  following  simple  Construction  of  the 
I.  Image- Point  S'  corresponding  to  an  Object-Point  5  on  the  chief 
incident  ray  u : 

Having  constructed  the  refracted  chief  ray  u'  corresponding  to  the 
chief  incident  ray  u,  draw  CY  perpendicular  to  u  at  F  and  YK 
perpendicular  to  CZ  at  K-,  the  straight  line  connecting  5  with  K 
will  intersect  the  chief  refracted  ray  u'  in  the  I.  Image-Point  S' '. 

The  position  of  the  Centre  of  Perspective  K  may  also  be  computed 
as  follows: 

Since 

CY  =  r-sin  a,      CK  =  CT-sin  a'  =  r-sin  a -sin  a', 

we  obtain: 

nr-sin2a      n'r-sin2a'  ,       x 

CK  =  -^~       — -•  (242) 

If  we  draw  the  straight  line  BK  (Fig.  127),  then 

Z  CBK  =  Z  CBZ'  -  Z.KBZ'  =  a'  -  ZjOZ' 

=  «'  -  (LBKC-  Z£Z'C)  =a+a'  - 

thus,  in  the  triangle  BKCwe  obtain: 

BC_  _  sin  Z.BKC sin 

CK  ~  sin  Z  CBK  ~  sin  («  + 

1  THOMAS  YOUNG:  On  the  Mechanism  of  the  Human  Eye:   Phil.  Trans.,  1801,  xcii.' 
p.  23.     This  paper  is  reprinted  in  The  Works  of  THOMAS  YOUNG,  in  three  volumes,  edited 
by  GEO.  PEACOCK,  D.D.  (London,  JOHN  MURRAY,  1855);  Vol.  I,  pages  12-63.    See  "Prop. 
IV."  on  p.   16. 

2  A.  CORNU:  Caustiques  —  Centre  de  Jonction:   Nouv.  Ann.de  Math.,  1863,  (2),  ii.t 
311-317.     See  also  A.  CORNU:  Construction  geometrique  des  deux  images  d'un  point 
lumineux  produit  par  refraction  oblique  sur  une  surface  spherique;  Journ.  de  physique, 
Ser.  Ill,  x.  (1901),  607. 

3F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme  an 
Kugelflaechen :  Wiener  Denkscht.,  1878,  xxxviii.,  163-192. 


340  Geometrical  Optics,  Chapter  XI.  [  §  235. 

And  since 


YC  CK 

=  sm  a,     -        =  sm  a', 


we  have  also: 


CK      sin  a  •  sin  a'  ' 
so  that 

i  sin  £BKC 


sin  a  •  sin  a'       sin  (a  +  a'  - 
whence  we  find  : 

tan  Z  BKC  =  tan  a  +  tan  a'.  (243) 

Commenting  on  this  result,  we  observe  that  /.BKC,  and,  hence,  also 
Z  C£  ^  =  a  +  a'  —  Z  B  K  C,  is  independent  of  the  radius  of  the 
spherical  refracting  surface,  so  that  the  values  of  these  angles  will 
depend  only  on  the  angle  of  incidence  a  and  on  the  indices  of  refrac- 
tion w,  n' . 

Indeed,  it  is  obvious  also  from  the  geometrical  relations  in  the  dia- 
gram, that  if,  keeping  the  incidence-angle  a  unchanged,  we  suppose 
the  radius  of  the  refracting  sphere  to  be  variable,  although  the  actual 
distances  from  B  of  both  C  and  K  will  vary,  the  directions  of  the 
straight  lines  B  C  and  B  K  will  remain  unaltered.  Perhaps,  the  easiest 
way  of  seeing  this  is  by  drawing  through  any  point  on  B  C  a  straight 
line  parallel  to  CZ,  and  constructing  a  point  on  this  line  exactly  in 
the  same  way  as  the  point  K  was  constructed  on  CZ.  It  will  be  seen 
that  the  point  thus  determined  will  lie  always  on  the  straight  line  B  K. 

235.    The  Focal  Points  J  and  /'  of  the  Meridian  Rays. 

If  the  Object-Point  S  is  the  infinitely  distant  point  /  of  the  chief 
incident  ray  u,  the  meridian  incident  rays  will  be  a  pencil  of  parallel 
rays  to  which  will  correspond  a  pencil  of  meridian  refracted  rays  meet- 
ing the  chief  refracted  ray  u'  in  the  "  Flucht  "  Point  /'  of  the  range 
of  I.  Image-Points.  And,  on  the  other  hand,  if  the  I.  Image-Point 
S'  is  the  infinitely  distant  point  /'  of  the  chief  refracted  ray  u' ,  the 
meridian  incident  rays  will  intersect  in  the  "  Flucht "  Point  /  of  the 
range  of  I.  Object-Points  lying  on  the  chief  incident  ray  u.  The 
"  Flucht "  Points  /  and  /',  or,  as  we  shall  now  call  them,  the  Primary 
and  Secondary  Focal  Points  of  the  Meridian  Rays,  may  be  located  by 
drawing  through  K  (Fig.  128)  straight  lines  parallel  to  u'  and  u  meet- 
ing u  and  u'  in  the  points  /  and  /',  respectively. 

According  to  a  well-known  law  of  projective  ranges  of  points,  we 
have  evidently: 

JB-I'B  =  JS'I'S'  =  (JB  +  BS)  (I'B  +  BS')-, 


§  235.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  341 

and  if  we  put 

BS  =  s,     BS'  =  s', 

we  obtain  the  following  equation : 

BJ       BIr 

V^    =I'  (244) 

which,  as  the  reader  will  remark,  is  completely  analogous  to  the 
relation 

AF/u  +  AE'lu'  =  i,     or    f/u  +  e'/u'  =  -  i, 

which  we  found  for  the  case  of  an  infinitely  narrow  bundle  of  normally 
incident  rays  refracted  at  a  spherical  surface;  see  formulae  (148). 


FIG.  128. 

REFRACTION  AT  A  SPHERICAL  SURFACE  OF  AN  INFINITELY  NARROW  BUNDLE  OF  RAYS.  Per- 
spective Relations  of  the  Range  of  Object- Points  lying  on  the  chief  incident  ray  u  and  the  Ranges 
of  I.  and  II.  Image-Points  lying  on  the  corresponding  refracted  ray  u'. 

The  positions  of  the  Focal  Points  /,  /'  may  be  calculated  as  follows  : 
From  the  figure  (Fig.  128),  we  obtain: 


BJ  =  I'K  =  -KZ'-- 


sin  (a  —  a') 


=  -CZ'-  S 


sin  (a  —  a') 


f\  ' 


and  since 


BI'  =  JK  =  KZ  •  -.    f"'    ,.  =  CZ-  S«^^S 
sin  (a  —  a)  sin  (a  —  a) 


CZ  =  n'rfn,      CZ'  =  nr/n', 


342  Geometrical  Optics,  Chapter  XI.  [  §  237. 

the  above  formulae  may  be  written: 

r  -  sin  OL  •  cos2  a  nr  -  cos2  a 

nj  —  — 


sin  (a  —  a') 

•      (HS) 


r  •  sin  a  •  cos2  a'  n'r  •  cos2  a! 

til    = 


sin  (a  —  a') 

236.    Formula  for  Calculating  the  Position  of  the  I.  Image-Point 
Sr  corresponding  to  an  Object-Point  5  on  a  given  incident  chief  ray  u. 

If  in  formula  (244)  above  we  substitute  the  values  of  BJ  and  BI' ', 
as  given  by  formulae  (245),  we  obtain  the  following  relation: 


n'  -  cos2  a'       n  •  cos2  a 


(246) 


Thus,  if  the  chief  incident  ray  u  is  given,  and  if  the  corresponding 
chief  refracted  ray  u'  has  been  calculated  trigonometrically,  so  that 
the  values  of  both  a  and  a  are  known,  this  useful  formula  enables  us 
to  calculate  the  value  of  sf  in  terms  of  that  of  5.  It  should  not  be 
forgotten,  however,  that  this  formula  has  been  obtained  by  neglecting 
magnitudes  of  the  second  order  of  smallness,  and  is  correct,  therefore, 
only  to  that  degree  of  approximation.  The  formula  may  be  written 
in  ABBE'S  differential  system  of  notation  (§  126)  as  follows: 

A  (  -  -  1  =  —  A(w-cosa).  (246*1) 

This  formula  may  be  derived  also  without  much  difficulty  from 
formulae  (191)  of  Chap.  IX  by  regarding  X,  X'  and  the  differences 
(a  —  a)  ,  (a'  —  a')  all  as  small  magnitudes  whose  second  powers  may  be 
neglected.  RAYLEIGH1  has  obtained  the  formula  also  in  a  very  simple 
way  by  the  use  of  the  Principle  of  the  Shortest  Light-  Path  (§38). 

237.     Convergence-Ratio  of  Meridian  Rays. 
s    If  (Fig.  127)  we  put 

Z  BSG  =  dX,     Z  BS'G  =  d\', 
then 


will  denote  the  convergence-ratio  of  the  pencil  of  meridian  rays.     Now 

*J.  W.  STRUTT,  Lord  RAYLEIGH:    Investigations  in  Optics  with  special  reference  to 
the  Spectroscope:   Phil.  Mag.,  (5),  ix.  (1880),  4O-S5- 


§  239.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  343 

we  saw  above  that 

BY  BY' 


and,  accordingly: 

BS 


__ 

/.BSG  "BY  '  BS'' 
Hence,  since 

BY  =  2r-cosa,     BY'  =  2r-cos«', 
we  obtain: 

d\'      s  •  cos  a' 


ART.  72.     THE  SAGITTAL  RAYS. 

238._  Relation  between  the  Object-Point  3  and  the  II.  Image- 
Point  3'. 

Let  S  designate  the  vertex  of  the  pencil  of  sagittal  incident  rays. 
If  the  bundle  of  incident  raysjs  homocentric,  ~S  will  coincide  with  S. 
We  have  seen  that  the  vertex  S'  of  the  pencil  of  sagittal  refracted  rays 
is  the  point  of  intersection  of  the  chief  refracted  ray  u'  with  the  central 
line  SC;  and,  hence,  without  further  study,  we  may  make  the  fol- 
lowing statement: 

The  range  of  Object-  Points  P,  Q,  R,  S,  •  •  •  lying  on  the  incident 
chief  ray  u  is  in  perspective  with  the  corresponding  range  of  II.  Image- 
Points  P',  Q',  R',  S',  -  -  -  lying  on  the  chief  refracted  ray  u'  ;  the  centre 
C  of  the  spherical  refracting  surface  being  the  Centre  of  Perspective  of 
these  two  corresponding  ranges,  since  the  straight  lines  PP',  QQ',  RR', 
SS',  •  -  •  all  pass  through  the  centre  C. 

239.    The  Focal  Points  J,  T  of  the  Sagittal  Rays. 

If  the  Object-Point  S  is  the  infinitely  distant  point  I  (or  /)  of  the 
chief  incident  ray  u,  the  sagittal  incident  rays  will  be  a  pencil  of 
parallel  rays  to  which  will  correspond  a  pencil  of  sagittal  refracted 
rays  with  its  vertex  at  the  "Flucht"  Point  T  of  the  range  of  II.  Image- 
Points  lying  on  the  chief  refracted  ray  u';  and,  similarly,  if  the  II. 
Image-Point  S'  coincides  with  the  infinitely  distant  point  J'  (or  /) 
of  the  chief  refracted  ray  u',  the  sagittal  refracted  rays  will  be  a  pencil 
of  parallel  rays  to  which  will  correspond  a  pencil  of  sagittal  incident 
rays  with  its  vertex  at  the  "Flucht"  Point  J  of  the  range  of  II. 
Object-Points  lying  on  the  chief  incident  ray  u. 

The  positions  of  the  "Flucht"  Points  /  and  /',  or,  as  we  shall  now 
call  them,  the  Primary  and  Secondary  Focal  Points  of  the  Sagittal  Rays, 


344  Geometrical  Optics,  Chapter  XI.  [  §  240. 

may  be  found  by  drawing  through  C  straight  lines  parallel  to  u'  and 
u  which  will  meet  u  and  u'  in  the  points  J  and  T,  respectively. 

Since  the  ranges  P,  Q,H.,S,  ••  •  and  ?',  Q',  R',  S',  •  •  •  are  pro- 
jective,  and  since  the  point  B  is  a  double-point  of  the  two  ranges,  we 
have  the  following  relation : 

JB  •  TB  =  JS-TS'  =  (JB  +  BS)(I'B  +  BS') ; 

and,  hence,  if  we  put 

BS  =  s,     BS'  =  s', 

we  derive  an  equation  exactly  analogous  to  formula  (244)  which  was 
obtained  for  the  meridian  rays,  viz. : 

BJ      BT 

T+-=T=i.  (248) 

The  positions  of  the  Focal  Points  J,  /'  may  be  calculated  as  follows : 
Since 

=  TC,     BI'  =  JC, 


we  obtain  directly  (Fig.  128): 

r  •  sin  a'  nr 


sin  (a  —  a')          n'  -  cos  a'  —  n  •  cos  a 

(249) 

BT  =         r'Sma      = — 

sin  (a  —  a')          n'  -  cos  a'  —  n  •  cos  a  ' 

240.  Formula  for  Calculating  the  Position  of  the  II.  Image-Point 
5'  corresponding  to  an  Object-Point  5  on  a  given  chief  incident  ray  u. 

Substituting  in  formula  (248)  the  values  of  BJ  and  BT,  as  given 
by  formula  (249),  we  obtain  the  following  formula  for  determining  s' 
in  terms  of  s: 


(250) 


or,  in  ABBE'S  method  of  writing : 

(n  \ 
-)=  -A(w-cosa).  (2500) 

S  /         T 

This  formula,  which  enables  us,  for  given  values  of  r,  a  and  a',  to  cal- 
culate the  position  of  the  II.  Image-Point  S'  on  u'  corresponding  to  a 
given  Object-Point  S  on  u,  remains  true  even  if  we  retain  infinitesimals 
of  the  second  order;  whereas  the  corresponding  formula  (246),  which 
we  obtained  for  the  meridian  rays,  is  only  correct  so  long  as  we  neglect 


§  242.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  345 

infinitesimals  of  the  second  order.  If  we  retained  infinitesimals  of  the 
second  order,  the  formula  which  would  be  obtained  for  the  meridian 
rays  would  give  values  of  s'  which  depend  on  the  inclinations  of  the 
secondary  rays  to  the  chief  ray  of  the  pencil  of  meridian  rays:  that 
is,  the  values  of  sf  would  differ  from  each  other  by  infinitesimals  of 
the  first  order.  Thus,  as  has  been  stated  above  (§  231),  the  conver- 
gence of  the  meridian  rays  at  the  I.  Image-Point  is  a  "convergence  of 
the  first  order",  whereas  the  convergence  of  the  sagittal  rays  at  the 
II.  Image-Point  is  a  "convergence  of  the  second  order". 

241.  Convergence-Ratio  of  the  Sagittal  Rays. 

Let  d\,  d\'  denote  the  angular  apertures  of  the  pencils  of  sagittal 
incident  and  refracted  rays.  Obviously,  we  have  the  following  re- 
lation : 

'          2«  =  f  =  f:  050 

where  Zu  denotes  the  Convergence-Ratio  of  the  Sagittal  Rays. 

ART.  73.       THE    ASTIGMATIC    DIFFERENCE,    AND    THE    MEASURE    OF   THE 

ASTIGMATISM. 

242.  If  the  bundle  of  incident  rays  is  homocentric,  the  Object- 
Points  5  and  S  on  the  chief  incident  ray  u  are  coincident,  and  in  this 
case,  therefore,  we  shall  have  s  =  s.     Thus, 

To  a  range  of  Object- Points  P,  Q,  R,  5,  •  •  •  lying  on  the  chief  inci- 
dent ray  u  there  corresponds  a  projective  range  of  I.  Image- Points  P',  Q', 
R',  S',  •  •  -  and  a  projective  range  of  II.  Image- Points  P',  ~Q',  #',  5', 
•  •  • ,  both  lying  on  the  chief  refracted  ray  u' ;  and,  hence,  also,  the  two 
ranges  of  Image- Points  are  projective  with  each  other. 

The  points  designated  in  the  figures  by  the  letters  B  and  Z'  are 
the  double-points  of  these  two  projective  ranges  of  Image-Points.  At 
the  incidence-point  B  the  Object-Point  and  its  two  Image-Points  coin- 
cide. The  point  Z',  as  we  saw,  is  the  vertex  of  the  bundle  of  refracted 
rays  corresponding  to  a  homocentric  bundle  of  incident  rays — which 
need  not  be  an  infinitely  narrow  bundle — with  its  vertex  at  the  Object- 
Point  Z  where  the  chief  incident  ray  meets  the  auxiliary  spherical 
surface  r  (see  §  207). 

In  the  case  of  an  infinitely  narrow  homocentric  bundle  of  Object- 
Rays  proceeding  from  the  Object-Point  51  and  undergoing  refraction 
at  a  spherical  surface,  the  astigmatic  difference  is  the  segment 

S'S'  =  s'  -  s' 
of  the  chief  refracted  ray  u'  comprised  between  the  II.  Image-Point 


346  Geometrical  Optics,  Chapter  XI.  [  §  242. 

and  the  I.  Image-Point.  The  astigmatic  difference  may  also  be  defined 
as  the  central  projection  from  the  homocentric  object-point  5  of  the 
line-segment  CK  on  the  chief  refracted  ray  u'. 

However,  as  the  Measure  of  the  Astigmatism  of  the  astigmatic  bundle 
of  refracted  rays,  it  is  found  more  convenient  to  take,  not  the  length 
of  the  segment  of  the  chief  refracted  ray  comprised  between  the  two 
Image-Points,  but  the  value  of  the  expression 


Thus,  let  us  suppose  that  the  bundle  of  incident  rays  is  not  homo- 
centric,  but  that,  in  consequence,  perhaps,  of  previous  refractions,  it 
also  is  an  astigmatic  bundle.  In  this  general  case  the  magnitudes 
denoted  by  5  and  s  will  not  be  equal.  Combining  equations  (246) 
and  (250),  and  using  the  relations 


we  obtain: 


cos2  a  =  I  —  sin2  a,     cos2  OL    =  I  —  sin2  a', 


2  OL       n  •  sin2  ot 


which  equation,  by  introducing  the  so-called  "Optical  Invariant" 

K  =  n  •  sin  a  =  n'  •  sin  a', 
may  be  written  in  ABBE'S  method  of  notation  as  follows: 


This  formula  gives  the  expression  for  the  measure  of  the  Change  of 
Astigmatism  produced  by  the  refraction  of  the  bundle  of  rays  at  the 
spherical  surface. 

In  particular,  we  may  observe  that  in  the  special  case  when  we  have 
n's'  =  ns,  the  change  of  astigmatism  is  equal  to  zero.  Thus,  for 
example,  if  the  bundle  of  incident  rays  is  homocentric  (s  =  s),  the 
bundle  of  refracted  rays  will  be  homocentric  also  (s'  =  s'),  provided 
n's'  =  ns.  We  have  seen  that  Z'  is  the  homocentric  Image-Point  of 
the  Object-Point  Z;  and,  therefore,  we  must  have  n'-BZ'  =  n-BZ 
(see  §  207).  Thus,  for  the  pair  of  points  Z,  Z'  we  obtain  from  equa- 
tion (246)  : 

BZ  =  r(cos  a.  +  n'  -cos  a'/n),^, 

BZf  =  r(n-cosa/n'  +  cosa').J 


§  243.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  347 

The  points  Z,  Z'  are  the  so-called  aplanatic  points  of  the  spherical 
refracting  surface. 

ART.  74.     HISTORICAL  NOTE,  CONCERNING  ASTIGMATISM. 

243.  The  theory  of  Astigmatism,  at  least  in  its  beginnings  and 
early  development,  is  due  almost  entirely  to  British  men  of  science. 

The  earliest  investigations  along  this  line,  of  which  we  have  any 
record,  are  to  be  found  in  the  optical  writings  of  the  distinguished 
mathematician  ISAAC  BARROW,  professor  of  Geometry  in  the  Univer- 
sity of  Cambridge  (1663-1669)  and  the  preceptor  of  NEWTON,  who 
succeeded  to  his  chair  in  the  university,  and  who  aided  BARROW  in 
preparing  for  publication  his  Lectiones  Opticce  (London,  1674).  In 
this  excellent  and  interesting  work,  BARROW  investigates  very  skil- 
fully the  paths  of  rays  lying  in  the  meridian  plane  of  a  spherical 
refracting  surface,  and  shows  how  to  construct  the  I.  Image-Point. 

But  the  real  discoverer  of  Astigmatism  was  Sir  ISAAC  NEWTON 
himself,  who  in  his  Lectiones  Opticce  Annis  1669,  1670,  1671  (London, 
1728)  deals  with  the  problem  of  the  refraction  of  a  narrow  bundle  of 
rays  at  both  plane  and  spherical  surfaces,  and  who  not  only  recognizes 
the  existence  of  the  two  Image-Points,  but  seeks  also  to  determine  what 
intermediate  point  is  selected  by  the  eye  as  the  place  of  the  image. 

The  next  most  important  advances  in  this  study  were  made  by 
ROBERT  SMITH,  who  investigated  very  thoroughly  the  properties  of 
caustics  both  by  reflexion  and  by  refraction  at  spherical  surfaces,  and 
who  showed  clearly  the  relations  between  the  Object-Point  and  the 
I.  Image-Point,  not  merely  for  the  case  of  refraction  or  reflexion  at  a 
single  spihercal  surface,  but  for  the  general  case  of  refraction  through 
a  centered  system  of  spherical  surfaces.  See  especially  Chapter  IX 
of  Book  2  of  SMITH'S  Compleat  System  of  Opticks  (Cambridge,  1738). 
Thus,  in  Sec.  423  (Vol.  i.,  p.  165)  SMITH  finds  that 

JS-I'S'  =  JB-I'B, 

where  the  letters  here  used  refer  to  Fig.  128.  This  result  is  a  direct 
consequence  of  the  fact  that  if  »P',  Qf,  Rf,  Sf,  etc.,  lying  on  the  chief 
refracted  ray  u' ,  are  the  I.  Image-Points  of  P,  Q,  R,  5,  etc.,  respect- 
ively, lying  on  the  chief  incident  ray  u,  these  two  ranges  of  points, 
as  we  found  in  §  233,  are  protective  with  each  other,  so  that  we  have  : 

(PQRS)  =  (P'Q'R'S'}. 

For  example,  according  to  this  relation,  we  have  (Fig.  128)  : 
JS-I'S'  =  JY-I'Y'  =  JZ-I'Z'  =  JB-I'B  =  a  constant. 


348  Geometrical  Optics,  Chapter  XI.  [  §  243. 

For  the  Construction  of  the  Focal  Points  J,  I',  SMITH  gives  (Sec. 
419,  Vol.  i.,  p.  164)  the  following  convenient  method:  From  the 
centre  C  (Fig.  128)  of  the  spherical  refracting  surface  draw  CY,  CY' 
perpendicular  at  F,  Yf  to  the  chief  incident  and  refracted  rays  u,  u1 \ 
respectively;  and  draw  the  radius  CB  to  the  point  of  incidence  B. 
From  F,  Yf  drop  perpendiculars  on  CB,  and  through  the  foot  of  the 
perpendicular  let  fall  from  F  draw  a  straight  line  parallel  to  u' ,  and 
through  the  foot  of  the  perpendicular  let  fall  from  Y'  draw  a  straight 
line  parallel  to  u.  These  straight  lines  will  intersect  u  and  u'  in  the 
required  points  /  and  /',  respectively. 

Among  the  most  important  contributions  to  this  subject  are  those 
of  THOMAS  YOUNG,  who  recognized  clearly  and  distinctly  the  value 
of  NEWTON'S  discovery  of  Astigmatism.  In  YOUNG'S  celebrated  paper 
"On  the  mechanism  of  the  eye"  (Phil.  Trans.,  1801,  cii.,  23-88;  re- 
printed in  the  Miscellaneous  Works  of  the  late  THOMAS  YOUNG,  London, 
1855),  he  gives  the  formula,  obtained  first  by  L'HOSPITAL  ("Analyse 
des  infiniment  petits" ',  Second  Edition,  Paris,  1716),  for  calculating 
the  intercept  s'  on  the  chief  refracted  ray  of  the  meridian  rays,  and 
shows  how  to  find  the  positions  of  the  Focal  Points  /',  /'  of  both  the 
meridian  and  the  sagittal  rays.  Moreover,  YOUNG  perceived  the  per- 
spective centres  K  and  C  of  the  rays  of  the  meridian  and  sagittal  sect- 
ions of  the  narrow  bundle  of  rays,  and  also  discussed  very  thoroughly 
the  astigmatism  of  the  eye.  In  his  Lectures  on  Natural  Philosophy 
(London,  1807),  YOUNG  gives,  likewise,  the  formula  for  the  intercept 
s'  of  the  sagittal  refracted  rays  and  applies  all  these  various  formulae 
to  a  number  of  important  special  cases.  He  seems  also  to  have  been 
the  first  to  recognize  the  existence  of  "image-lines".  Moreover, 
YOUNG  was  cognizant  of  the  so-called  "aplanatic"  points  of  a  refract- 
ing sphere. 

The  contributions  of  AIRY  *  and  of  CODDINGTON  2  to  the  theory  of 
astigmatism  deserve  also  to  be  ranked  among  the  most  important. 

For  a  complete  and  very  learned  account  of  the  theory  of  Astigma- 
tism from  the  earliest  times  to  the  present,  the  reader  is  referred  to 
the  historical  note,  "Ueber  den  Asfigmatismus",  at  the  end  of  P. 
CULMANN'S  article  entitled  "Die  Realisierung  der  optischen  Abbild- 
ung",  which  is  Chapter  IV  of  Die  Theorie  der  optischen  Instrument, 
edited  by  M.  VON  ROHR  (Berlin,  1904). 

1  G.  B.  AIRY:  On  a  peculiar  defect  in  the  eye  and  mode  of  correcting  it:   Camb.  Phil. 
Trans.  (1827),  ii.,  227-252.     Also:  On  the  spherical  aberration  of  the  eye-pieces  of  tele- 
scopes (Cambridge,  1827);  and  in  Camb.  Phil.  Trans.,  iii.  (1830),  1-64. 

2  H.  CODDINGTON:  A  Treatise  on  the  Reflexion  and  Refraction  of  Light:  London,  1829. 


§  244.]        Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  349 

ART.  75.     INQUIRY  AS  TO  THE  NATURE  AND  POSITION  OF  THE  IMAGE  OF 
AN  EXTENDED  OBJECT  FORMED  BY  NARROW  ASTIG- 
MATIC BUNDLES  OF  RAYS. 

244.  It  appears,  therefore,  that,  in  general,  when  an  infinitely 
narrow  homocentric  bundle  of  rays  is  refracted  at  a  spherical  surface, 
the  bundle  of  refracted  rays  will  not  be  homocentric,  but  will  be  astig- 
matic; so  that  to  an  Object-Point  there  corresponds,  not  a  single 
Image-Point,  such  as  we  have  in  the  case  of  ideal  imagery,  but  a  pair 
of  infinitely  short  Image-Lines  at  right  angles  to  each  other  and  lying 
in  different  planes.  An  eye  placed  on  the  chief  refracted  ray  may 
accommodate  itself  to  regard  either  of  these  two  Image-Lines  as  the 
image  of  the  Object-Point  whence  the  rays  emanate. 

If,  instead  of  one  single  Object-Point,  we  have  an  aggregation  of 
such  points  forming  an  extended  object,  each  of  these  points  being  the 
vertex  of  an  infinitely  narrow  bundle  of  incident  rays  whose  chief  rays 
(we  may  suppose)  all  meet  the  spherical  refracting  surface  at  the  same 
point  B,  the  image  of  the  object  will  be  more  or  less  blurred  and 
distorted.  Thus,  if  the  eye  is  accommodated  to  view  the  primary 
Image-Lines,  the  dimensions  of  the  object  parallel  to  these  lines  will 
be  exaggerated  in  the  image,  whereas  when  the  eye  is  focussed  on 
the  other  set  of  Image-Lines,  there  will  be  a  similar  exaggeration 
parallel  to  these  lines,  so  that  in  either  case  the  quality  of  the  image 
will  be  defective.  Thus,  as  a  rule,  we  do  not  obtain  either  faithful  or 
distinct  images  by  means  of  astigmatic  bundles  of  rays.  It  is  assumed 
by  most  writers  that  on  the  whole  the  best  image  in  such  a  case  will 
be  obtained  by  accommodating  the  eye  to  view  neither  of  the  two  sets 
of  Image-Lines  of  the  astigmatic  bundle  of  rays,  but  a  place  lying 
somewhere  between  these,  the  place  of  the  so-called  "Circle  of  Least 
Confusion1'.  In  fact,  it  is  said,  the  eye  unconsciously  selects  these 
sections  of  the  astigmatic  bundles  of  rays.1  Correspondingto  each 
point  of  the  object,  the  eye  will  thus  see  a  small  area,  so  that  according 
to  this  view  of  the  matter,  the  image  of  an  object,  as  HEATH  expresses 
it,  "is  taken  to  be  the  aggregation  of  the  overlapping  'Circles  of  least 
confusion'."  In  general,  this  is  no  doubt  a  correct  explanation,  but 
in  some  special  cases  a  more  perfect  and  satisfactory  image  may  be 
obtained  by  viewing  the  Image-Lines  directly. 
,  CzAPSKi2  considers,  for  example,  the  case  of  an  infinitely  short 

1  See,  for  example,  HEATH'S  Geometrical  Optics  (Cambridge,  1887),  Art.  145.     Also, 
O.  LUMMER'S   work  on  Optics,    published   as  Vol.  II  of  the  Ninth  Edition  of  MUELLER- 
POUILLET'S    Lehrbuch  der  Physik,  Art.   183.  —  The  designation    of  this  section  of  the 
bundle  of  rays  as  the  place  of  "least  confusion"  is  rather  misleading,  as  the  definition  is 
better  at  either  of  the  two  Image-Lines. 

2  S.  CZAPSKI  :  Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  S.  76.     See. 


350 


Geometrical  Optics,  Chapter  XI. 


§244. 


Object-Line  a  perpendicular,  say,  at  5  to  the  incident  ray  SB  (or  u) 
which  we  may  consider  here  as  a  "mean"  incident  chief  ray.  From 
each  point  of  the  Object-Line  a  there  proceeds  a  bundle  of  object-rays 
which  all  meet  the  spherical  refracting  surface  in  points  closely  adjacent 
to  the  incidence-point  B  of  the  "mean"  incident  chief  ray.  We 
shall  regard  as  the  chief  rays  of  all  these  bundles  of  rays  those  rays 
which  meet  the  refracting  surface  at  the  point  B.  Thus,  as  CZAPSKI 
says,  the  conditions  are  very  nearly  the  same  for  all  these  bundles  of 
object-rays,  and,  therefore,  we  shall  have  very  nearly  the  same  phe- 
nomena. It  is  true  that  the  chief  rays  of  the  bundles  will  meet  the 
spherical  refracting  surface  at  slightly  different  angles  of  incidence, 
and,  consequently,  the  astigmatic  differences  of  the  corresponding 
bundles  of  refracted  rays  will  be  also  slightly  different;  but  the  I. 
Image-Lines  will  all  be  very  nearly  parallel,  and  the  same  is  true  also 
of  the  II.  Image-Lines.  Moreover,  the  lengths  of  the  lines  in  each 
group  will  not  be  very  different  from  each  other. 

The  image  of  the  Object-Line  a  will  appear,  therefore,  as  the  as- 
semblage of  the  I.  Image-Lines  a'  (Fig.  129)  or  of  the  II.  Image-Lines 
a'  (Fig.  1 29) ,  according  as  the  eye  is  focussed  to  view  one  or  the  other 

of  these  aggregations.  In  either  case,  the 
image  will  evidently  be  an  unsatisfactory 
representation  of  the  object. 

But  if  the  Object-Line  is  a  short  line  b 
lying  in  the  plane  of  incidence  of  the  "mean" 
chief  ray  (Meridian  Plane),  and  perpendicu- 
lar to  this  ray,  we  have  a  special  case  that 
is  worth  considering.  Now  all  the  chief 
rays  of  the  bundles  of  object-rays  will  lie 
in  the  Meridian  Plane,  so  that  the  I.  Im- 
age-Points S'  and  the  II.  Image-Points  3' 
corresponding  to  the  points  of  the  Object- 
Line  b  all  lie  in  the  plane  of  incidence.  The 
I.  Image-Lines  are  perpendicular  to  the 
plane  of  incidence  at  the  I.  Image-Points, 
so  that  the  image  produced  by  this  assem- 
blage of  Image-Lines  will  have  the  form  of 
a  rectangle  b'  (Fig.  129)  perpendicular  to  the  plane  of  incidence.  The 
II.  Image-Lines,  on  the  other  hand,  lie  in  the  plane  of  incidence,  being 

also,  P.  CULMANN'S  article  "  Die  Realisierung  der  optischen  Abbildung  ",  which  forms 
Chapter  IV  of  Theorie  der  optischen  Instrumente,  edited  by  M.  VON  ROHR  (Bd.  I,  Berlin, 
1904),  S.  167.  Also,  Theorie  und  Geschichte  des  Photographischen  Objektivs,  by  M.  VON 
ROHR  (Berlin,  1899),  42,  43. 


111!!! 


FIG.  129. 

IMAGES  OF  A  SMALL  OBJECT- 
lyiNE  PERPENDICULAR  TO  THE 
"MEAN"  CHIEF  INCIDENT  RAY, 
as  produced  by  means  of  Narrow 
Astigmatic  Bundles  of  Refracted 
Rays. 


§  245.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  351 

spread  over  a  small  area  in  this  plane,  so  that  if  a  screen  were  placed 
at  right  angles  to  the  "mean"  refracted  chief  ray  at  S',  it  would  inter- 
sect the  bundles  of  refracted  rays  in  a  line.  Thus,  an  eye  placed  on 
the  "mean"  refracted  chief  ray  and  accommodated  to  the  II.  Image- 
Point  S'  would  view  there  a  linear  image  b'  (Fig.  129)  of  the  linear 
object  b. 

An  entirely  analogous  case  is  presented  when  the  Object-Line  is 
a  small  line  c  lying  in  the  plane  containing  the  "mean"  incident  chief 
ray  and  perpendicular  to  the  plane  of  incidence.  An  eye  placed  on  the 
"mean"  refracted  chief  ray  and  accommodated  for  the  I.  Image-Point  Sf 
would  see  there  an  image  of  c  in  the  form  of  a  straight  line  c'  (Fig.  129) 
parallel  to  c  itself;  whereas  if  the  eye  were  focussed  on  the  II.  Image- 
Point  3',  the  image  of  c  will  be  found  to  be  a  rectangular  figure  cf 
(Fig.  129)  perpendicular  at  S'  to  the  plane  of  incidence.1 

ART.  76.  COLLINEAR  RELATIONS  IN  THE  CASE  OF  THE  REFRACTION  OF 
A  NARROW  BUNDLE  OF  RAYS  AT  A  SPHERICAL  SURFACE. 

245.  The  Principal  Axes  of  the  Two  Pairs  of  Collinear  Plane 
Systems. 

To  the  chief  ray  u  of  an  infinitely  narrow  homocentric  bundle  of 
incident  rays  which  meets  the  spherical  refracting  surface  at  the  inci- 
dence-point B  corresponds  the  chief  refracted  ray  u'  of  the  astigmatic 
bundle  of  refracted  rays.  Both  the  incident  meridian  rays  and  the 
refracted  meridian  rays  proceed  in  the  plane  uu' ,  which  may,  therefore, 
be  designated  as  the  plane  TT  or  TT'.  Similarly,  the  planes  of  the  inci- 
dent sagittal  rays  and  the  refracted  sagittal  rays  may  be  designated  by 
the  symbols  TT,  TT',  respectively. 

Consider,  first,  a  point  V  lying  in  the  plane  TT  of  the  meridian  rays 
and  very  near  to  the  chief  incident  ray  u;  and  let  us  suppose  that  V 
itself,  regarded  as  an  Object-Point,  is  the  vertex  of  a  narrow  bundle 
of  incident  rays  all  meeting  the  spherical  surface  at  points  nearly 
adjacent  to  the  incidence-point  B.  The  incident  ray  VB  (or  v) 
lying  in  the  plane  TT  may  be  treated  as  the  chief  ray  of  this  bundle. 
The  angle  at  B  between  the  rays  u,  v  being  an  infinitesimal  angle  of 
the  first  order,  so  likewise  is  the  angle  between  the  corresponding  re- 
fracted rays  u' ,  v'\  and,  since  v  lies  in  the  plane  uC  or  TT,  v'  will  lie  in 
the  plane  u'  C  or  TT',  which  is  coincident  with  TT;  and  to  the  pencil  of 
incident  rays  proceeding  from  V  and  lying  in  the  plane  TT  will  corre- 

1  For  a  very  clear  and  interesting  treatment  of  the  images  formed  by  astigmatic  bundles 
of  rays  see  L.  MATTHIESSEN:  Ueber  die  Form  der  unendlich  duennen  astigmatischen 
Strahlenbuendel  und  ueber  die  KuMMER'schen  Modelle:  Sitzungber.  der  math.-phys.  Cl. 
der  k.  bayer.  Akad.  der  Wiss.  zu  Muenchen,  xiii.  (1883),  35-51- 


352  Geometrical  Optics,  Chapter  XL  [  §  245. 

spond  a  pencil  of  refracted  rays  lying  in  the  plane  TT'  and  converging  to 
the  I.  Image-Point  V  on  v'.  Thus,  if  we  utilize  only  such  rays  as 
before  and  after  refraction  proceed  infinitely  near  to  u  and  u' ,  respect- 
ively, the  plane-fields  TT,  TT'  will  be  characterized  by  the  property  that 
to  a  homocentric  pencil  of  rays  of  TT  there  corresponds  by  refraction  a 
homocentric  pencil  of  rays  of  TT'. 

In  the  next  place,  let  us  consider  a  point  W  lying  in  the  plane  of  the 
sagittal  section  and  also  infinitely  near  to  the  chief  ray  u  of  the  bundle 
of  incident  rays.  Regarding  W  as  an  Object-Point,  we  shall  suppose 
that  it  is  the  vertex  of  a  narrow  bundle  of  incident  rays  whose  chief  ray 
w  meets  the  spherical  refracting  surface  also  at  the  point  B  so  that 
the  angles  between  the  two  incident  chief  rays  u  and  w  and  between 
the  corresponding  refracted  chief  rays  u'  and  w'  are  both  infinitesimal 
angles  of  the  first  order.  If  we  use  YOUNG'S  Construction  (§  206)  for 
drawing  the  refracted  ray  w' ,  it  will  be  obvious  that,  if  we  neglect 
infinitesimals  of  the  second  order,  w'  will  lie  in  the  plane  TT';  and, 
with  the  same  degree  of  exactness,  all  incident  rays  proceeding  from 
W  and  lying  in  the  plane  TT  will,  after  refraction,  lie  in  the  plane  TT', 
and  will  converge  to  the  II.  I  mage- Point  W  on  w'  corresponding  to 
the  Object-Point  W  on  w. 

Thus,  within  the  infinitely  narrow  region  surrounding  the  so-called 
"mean"  incident  chief  ray  u  in  the  Object-Space  and  the  corresponding 
refracted  chief  ray  u'  in  the  Image-Space,  we  have  a  collinear  relation 
between  the  plane-fields  TT,  TT'  and  also  between  the  plane-fields  TT,  TT'; 
because  to  every  incident  ray  in  TT  (or  TT)  there  corresponds  a  refracted 
ray  in  IT'  (or  TT')»  and  to  every  Object-Point  of  TT  (or  ?r)  there  corre- 
sponds a  I.  (or  II.)  Image-Point  of  TT'  (or  ?r'). 

It  may  be  remarked  also  that  the  plane-fields  TT,  TT'  have  in  common 
the  range  of  points  which  lie  in  the  plane  of  incidence  TT  along  the 
tangent  to  the  spherical  refracting  surface  at  the  incidence-point  B\ 
whereas  the  plane-fields  TT,  TT'  have  in  common  the  range  of  points 
which  lie  in  the  line  of  intersection  of  these  planes. 

These  results,  which  appear  to  have  been  first  obtained  by  LiPPicn,1 
may,  accordingly,  be  stated  as  follows: 

(1)  The  plane-fields  TT,  TT'  lying  in  the  plane  of  incidence  are  in 
perspective  with  each  other;  and 

(2)  The  plane-fields  TT,   TT',  which  are  both   perpendicular  to  the 
plane  uur,  and  which  contain  u,  u',  respectively,  are  likewise  in  per- 
spective with  each  other. 

1 F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme 
an  Kugelflaechen:  Denkschriften  der  kaiserl.  Akad.  der  Wissenschaften  zu  Wien,  xxxviii. 
(1878),  163-192. 


§  245.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  353 

In  order  to  get  a  proper  idea  of  the  imagery  which  we  obtain  by  means 
of  the  meridian  rays,  suppose  we  consider  an  infinitely  short  object- 
line  SV  lying  in  the  plane  of  incidence  SB  C  and  perpendicular  at 
S  to  the  "mean"  incident  chief  ray  SB  (or  u).  To  the  narrow  pencil 
of  meridian  object-rays  with  its  vertex  at  V  will  correspond  a  pencil 
of  meridian  image-rays  with  its  vertex  at  the  I.  Image-Point  V"  on 
the  refracted  ray  BV"  corresponding  to  the  incident  chief  ray  VB\ 
and  if  S'  is  the  I.  Image-Point  on  u'  corresponding  to  the  Object-  Point 
S  on  u,  the  infinitely  short  line  S'V"  ,  which  is  the  image  of  the  object- 
line  SFwill,  in  general,  not  be  perpendicular  to  the  "mean"  refracted 
chief  ray  u'  (or  BSf).  Accordingly,  let  us  draw  S'Vr  perpendicular 
to  BS'  at  S'  and  meeting  BV"  in  the  point  V.  Now  the  distance 
between  the  two  points  V  and  V"  and  also  the  angular  aperture 
of  the  pencil  of  image-rays  V"  are  infinitesimals  of  the  first  order, 
and,  therefore,  the  piece  of  S'V  intercepted  between  the  two  extreme 
rays  of  this  pencil  will  be  an  infinitesimal  of  the  second  order,  and, 
consequently,  may  be  treated  as  a  mere  point,  since  here  we  neglect 
infinitesimals  of  order  higher  than  the  first.  Thus,  according  to  ABBE, 
we  may  regard  the  point  Vf  as  the  vertex  of  the  pencil  of  image-rays 
corresponding  to  object-rays  proceeding  from  V  and  S'V',  therefore, 
as  the  image  of  SV.  In  brief,  provided  we  neglect  infinitesimals  of 
the  second  order,  we  have  a  right  to  say  that  the  image,  by  means  of 
meridian  rays,  of  an  infinitely  short  object-line  perpendicular  to  the 
"mean"  incident  chief  ray  is  an  infinitely  short  line  perpendicular  to 
the  "mean"  refracted  chief  ray.1 

If  d\  denotes  the  inclination  to  the  chief  ray  u  of  a  secondary  ray 
of  the  pencil  of  meridian  object-rays  whose  vertex  is  at  5,  and  if  d\f 
denotes  the  inclination  to  the  chief  refracted  ray  u'  of  the  corresponding 
refracted  secondary  ray,  it  is  a  very  simple  matter  to  show  that 
(always  neglecting  infinitesimals  of  the  second  order)  we  have  for  the 
meridian  rays  the  following  relation: 


which  will  be  recognized  as  perfectly  analogous  to  the  Law  of  ROBERT 
SMITH  for  the  refraction  of  paraxial  rays,  the  so-called  LAGRANGE- 
HELMHOLTZ  Formula,  §  194. 

And,  finally,  if  we  consider  in  the  same  way  the  imagery  in  the 

1  See  CZAPSKI'S  Theorie  der  optischer  Instrumente  nach  ABBE  (Breslau,  1893),  S.  78. 
Also,  P.  CULMANN'S  "  Die  Realisierung  der  optischen  Abbildung  ",  which  forms  Chapter 
IV  of  Die  Theorie  der  optischen  Instrumente,  edited  by  M.  VON  ROHR  (Bd.  I,  Berlin,  1904), 
S.  171. 

24 


354  Geometrical  Optics,  Chapter  XI.  [  §  246. 

planes  TT,  TT',  it  will  be  obvious,  on  mere  grounds  of  symmetry,  that  the 
image,  by  means  of  the  sagittal  rays,  of  an  infinitely  short  object-line 
SW  lying  in  the  plane  TT  and  perpendicular  at  5  to  the  "mean"  incident 
chief  ray  u  will  be  an  infinitely  short  line  ~S'W  in  the  plane  TT'  and 
perpendicular  to  the  refracted  chief  ray  u'  at  the  II.  Image-Point  Sr 
corresponding  to  the  object-point  5;  provided  that  here  also  we  neglect 
infinitesimals  of  the  second  order. 

Thus,  according  to  ABBE,  the  "mean"  incident  chief  ray  u  and  the 
corresponding  refracted  ray  u'  are  to  be  regarded  as  the  Principal 
Axes  of  the  narrow  collinear  plane-fields  ?r,  TT'  and  also  of  the  narrow 
collinear  plane-fields  TT,  TT',  since  in  both  cases  to  an  object-line  perpen- 
dicular to  u  there  corresponds,  as, we  have  seen,  an  image-line  perpen- 
dicular to  u' .  This  was  not  the  case  in  LIPPICH'S  mode  of  treating 
this  matter,  but  it  will  be  found  to  simplify  the  problem  very  greatly 
to  be  able  to  consider  the  chief  rays  u,  u'  as  the  Principal  Axes  of  the 
two  pairs  of  collinear  plane  systems. 


FIG.  130. 

FIGURE  FOR  FINDING  THE  SECONDARY  FOCAL  LENGTH  (A/)  OF  THE  SYSTEM  OF  MERIDIAN 

RAYS. 


246.  Having  determined  the  Principal  Axes,  we  can  now  proceed 
to  obtain  the  formulae  for  calculating  The  Focal  Lengths  of  the  two 
plane  systems  of  rays;  the  Focal  Lengths  being  defined  as  in  §  178. 

For  example,  Fig.  130  represents  the  case  of  a  narrow  pencil  of 
parallel  meridian  incident  rays  to  which  corresponds  a  pencil  of  re- 
fracted meridian  rays  with  its  vertex  at  the  Focal  Point  If .  The 
incidence-points  of  the  chief  ray  u  and  a  secondary  ray  of  the  pencil 
of  incident  rays  are  designated  in  the  diagram  by  the  letters  B  and  Gy 
respectively;  the  corresponding  refracted  rays  are  El'  (or  u'}  and  GI'. 


§  246.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  355 

At  B  erect  BU,  BU'  perpendicular  to  u,  u'  and  meeting  the  secondary 
incident  ray  and  the  secondary  refracted  ray  in  the  points  U,  U', 
respectively.  If,  then,  we  put  d\'  =  Z  BI'G,  and  if  eu  denotes  the 
Secondary  Focal  Length  of  the  system  of  meridian  rays  for  which 
u  and  u'  are  the  chief  incident  and  refracted  rays,  according  to  the 
definition  referred-to  above,  we  shall  have: 


- 

u  ~  d\'  ' 

Similarly,  in  the  case  of  a  pencil  of  parallel  meridian  refracted  rays 
emanating  before  refraction  from  the  Focal  Point  /  on  the  chief  inci- 
dent ray  ut  the  Primary  Focal  Length  fu  will  be  given  by  the  formula  : 

BU' 

^M  ~    d\  ' 
where  d\  =  Z  B  JG. 

If  a,  «'  denote  the  angles  of  incidence  and  refraction  of  the  chief 
ray,  we  have  evidently  the  following  relations: 

BU  =  BG-cos  a,  BU'  =  BG-cos  a', 


BU  £G;cos  a  ,  _        BU'  _        BG-cos  a' 

BJ  "  BJ       '     ^X   :      "  BI'  ~~  BI' 

whence,  therefore,  we  obtain: 


cos  a'  cos  a 


fu  =  J  B  -—     —  ,     eu  =  I  B  •  —  —  ,  . 
cos  a  cos  a 

Thus,  we  see  that  the  Focal  Lengths  fu  and  e'u  are  not  equal  to  the 
segments  JB  and  I'  B  on  u  and  u'  comprised  between  the  Focal  Points 
/  and  /',  respectively,  and  the  incidence-point  B,  as,  having  in  mind 
the  special  case  of  normally  incident  rays,  where  we  have  /  =  FA  , 
e'  =  E'Aj  we  might  have  expected. 

Substituting  for  JB  and  I'  B  their  values  as  derived  from  formulae 
(245),  we  obtain  finally: 

nr  •  cos  a  -  cos  a'  ,  n'r  •  cos  a  -  cos  a.' 

u  ~  n'  -  cos  a'  —  n  •  cos  a  '     e"  ~        ri  -  cos  a'  —  n  •  cos  a  '        ^ 

By  a  process  exactly  similar  to  the  above,  we  shall  obtain,  even 
more  simply,  for  the  Focal  Lengths  /M,  eu  of  the  system  of  sagittal 
rays,  for  which  u  and  u'  are  the  chief  incident  and  refracted  rays, 
expressions  as  follows: 

L  =  JB,     e'H=I'B, 

so  that  in  the  case  of  the  sagittal  rays  the  distances  of  the  incidence- 


356  Geometrical  Optics,  Chapter  XI.  [  §  247. 

point  B  from  the  Focal  Points  J  and  I'  are  equal  to  the  Focal  Lengths. 
Thus,  employing  formulae  (249),  we  obtain: 

-  _  nr  ,  '          _  n'r 

n'-cosa'  —  n-cosa  '       "  w'-cos  a   —  w-cos  a  '         "' 

For  given  values  of  the  constants  n,  n'  and  r,  the  Focal  Lengths  /u,  eu 
and  /u,  eu,  as  we  see  from  formulae  (254)  and  (255),  depend  only  on 
the  angle  of  incidence  (that  is,  therefore,  on  the  slope  and  position) 
of  the  chief  incident  ray  u.  In  the  special  case  when  the  chief  inci- 
dent ray  meets  the  spherical  surface  normally  at  the  vertex  At  by 
putting  a  =  o,  in  the  formulae  (254)  and  (255),  and  writing  Fin  place 
of  /  or  J  and  Ef  in  place  of  /'  or  /'  and  A  in  place  of  B,  we  obtain: 

For  a  =  o:          /„  =/M  =/  =  FA  =  nr/(n'  -  n), 

e'  =  *'u  =eu  =  E'A  =  -n'r/(n'  -  n); 

as  in  formulae  (147)  of  Chapter  VIII. 
Moreover,  we  find  also: 

/«K  =/«/<=  -»/»',  (256) 

which  corresponds  with  the  relation  already  found  in  Chapter  VIII, 
viz.,f/e'  =  —  n/n'. 

The  magnification-ratios  for  the  meridian  and  sagittal  rays  may  be 
derived  without  difficulty  by  means  of  the  formulae  given  in  Chap. 
VII,  §  179- 

ART.  77.     REFRACTION   OF  NARROW  BUNDLE  OF  RAYS  THROUGH  A  CEN- 
TERED SYSTEM  OF  SPHERICAL  REFRACTING  SURFACES. 

247.  Formulae  for  Calculating  the  Astigmatism  of  the  Bundle  of 
Emergent  Rays. 

We  shall  consider  here  only  the  simple  case  when  the  chief  incident 
ray  u±  lies  in  a  plane  which  contains  the  optical  axis  of  the  centered 
system  of  spherical  surfaces.  Thus,  all  the  meridian  sections  of  the 
astigmatic  bundles  of  rays  arising  by  refraction  at  the  successive  sur- 
faces will  lie  in  this  plane. 

Let  Bk  designate  the  point  where  the  chief  ray  meets  the  &th  spheri- 
cal refracting  surface,  and  let 


denote  the  length  of  the  path  of  the  chief  ray  comprised  between  the 
incidence-point  Bk  at  the  fcth  surface  and  the  incidence-point  Bk+l 
at  the  (k  +  i)th  surface.  Moreover,  let  S'k,  Sk  designate  the  positions 
on  the  chief  ray  of  the  I.  and  II.  Image-Points,  respectively,  after 


§  247.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface. 


357 


the  refraction  of  the  ray  at  the  kth  surface. 
the  following  symbols: 

-   =  **» 


We  shall  employ  also 


BkS'k  = 


The  relations  between  these  intercepts  on  the  chief  ray  before  and 
after  refraction  at  the  kth  surface  are  given,  for  the  meridian  rays, 
by  formula  (246)  and,  for  the  sagittal  rays,  by  formula  (250)  of  this 
Chapter.  Thus  if  rk  (=  AkCk)  denotes  the  radius  of  the  kth  spherical 
surface,  and  if  ak,  <xk  denote  the  angles  of  incidence  and  refraction  of 
the  chief  ray  at  this  surface,  we  shall  have: 


n,.  •  cos  a 


nk_l  •  cos 


*»— 1 


%•  °k  °& 

where,  by  way  of  abbreviation,  we  have  put 


nt'C08ak-nk_l  -cos  o^ 
r^ 


(257) 


(258) 


this  magnitude  being  called  sometimes  the  "astigmatic  constant**  of 
the  kth  spherical  surface  for  the  ray  incident  on  it  at  the  angle  ak. 
For  the  Logarithmic  Computation  of  the  positions  on  the  emergent 
chief  ray  of  the  I.  and  II.  I  mage- Points  S'm  and  S'm  corresponding  to 
an  Object-Point  5X  on  the  chief  incident  ray,  it  will  be  necessary,  in 
the  first  place,  to  determine,  by  means  of  the  system  of  formulae  (215) 
of  Chapter  X,  the  path  of  the  chief  ray  through  the  centered  system 
of  m  spherical  refracting  surfaces,  whereby  we  shall  obtain  the  values 
of  the  angles  of  incidence  a,  a  at  each  surface  in  succession.  We  may 
then  proceed  to  employ  the  following  system  of  formulae,  which  are 
written  in  a  form  adapted  to  logarithmic  work: 


I.     Meridian  Rays: 


rk  •  sn  ak 


•  cos 


sk        nk  •  cos  ak     oj 

sk+i  =  sk  ~  ^k- 
II.     Sagittal  Rays : 

l.^fci.I   .  11*. 

^1    "   ^1      J*      »i  ' 


•  cos2  ak  ' 


(259) 


358  Geometrical  Optics,  Chapter  XI.  [  §  248. 

In  these  formulae  k  must  receive  in  succession  all  integral  values  from 
k=  I  to  k  =  m  (§m  =  o).  Accordingly,  if  we  are  given  the  values  of  the 
constants  of  the  optical  system,  that  is,  the  magnitudes  denoted  by  n,  r 
and  d,  and  if  we  are  also  given  the  ray-co-ordinates  (vlt  0J  of  the  chief 
ray  incident  on  the  first  spherical  surface,  so  that  we  have  the  data  for 
determining,  by  means  of  formulae  (215)  of  Chapter  X,  the  magnitudes 
denoted  by  a,  a  and  5;  and,  if  finally,  we  are  given  the  positions  on  the 
chief  incident  ray  of  the  I.  Object- Point  5t  and  of  the  II.  Object- 
Point  Slt  that  is,  if  we  are  given  the  values  of  the  intercepts  s^  =  B^) 
and  st  (=  .BjSi);  we  can,  by  successive  substitutions  in  formulae  (259), 
obtain  the  values  of  the  magnitudes  s'm  ( =  BmS'm)  and  s'm  ( =  BmS'm) ,  and 
thus  determine  the  positions  on  the  emergent  chief  ray  of  the  I.  and 
II.  Image-Points  S'm  and  S'm,  and  the  magnitude  of  the  Astigmatic 
Difference  S^S^  =  s'm  —  s'm.  The  calculation,  to  be  sure,  is  quite  long 
and  tedious,  especially  if  the  system  consists  of  as  many  as  four  or  five 
refracting  surfaces;  but  there  is  no  shorter  process  of  solving  the 
required  problem.1 

The  condition  that  the  Astigmatic  Difference  of  the  bundle  of  emer- 
gent rays  shall  vanish  is  S^S^  =  o,  or  s'm  =  "s'm.  If  the  Optical  System 
consists  of  a  single  Lens  (m  =  2),  it  is  not  difficult  to  show  that  this 
condition  leads  to  a  quadratic  equation  for  determining  sa(=  s^. 
The  problem  of  the  Homocentric  Refraction  of  Light-Rays  through 
a  Lens  has  been  beautifully  and  completely  investigated  by  L.  BUR- 
MESTER.2  By  a  simple  process  of  geometrical  reasoning,  he  shows 
that  when  an  infinitely  narrow  bundle  of  rays  is  refracted  through  a 
Lens,  there  are  two  object-points  (which  may  be  real  or  imaginary, 
and  which  may  be  coincident)  lying  on  the  chief  object-ray,  to  each 
of  which  there  corresponds  on  the  chief  image-ray  a  "Homocentric" 
Image-Point.  Moreover,  the  same  reasoning  can  be  extended  imme- 
diately to  show  that  the  same  thing  is  true  also  in  the  case  of  a  centered 
system  of  any  number  of  spherical  refracting  surfaces.  BURMESTER 
shows  also  how  to  construct  the  two  object-points  and  the  corre- 
sponding "Homocentric"  Image-Points  in  the  case  of  a  Lens,  and 
discusses  a  number  of  interesting  special  cases. 

248.     Collinear  Relations. 

Within  the  infinitely  narrow  region  surrounding  the  chief  ray  before 
and  after  refraction  at  the  kth  spherical  surface,  we  have  a  collinear 

1  See  A.  GLEICHEN:  Lehrbuch  der  geometrischen  Optik  (Leipzig  und  Berlin,  B.  G.  TEUB- 
NER,  1902),  pages  441-467,  for  the  complete  calculation  of  the  "Astigmatische  Bildpunkte" 
of  P.  GOERZ'S  Double  Anastigmatic  Photographic  Objective. 

8  L.  BURMESTER:  Homocentrische  Brechung  des  Lichtes  durch  die  Linse:  ZfL  /.  Math, 
u.  Phys.,  xl.  (1895),  321. 


§  248.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  359 

relation  between  the  plane-systems  TT^  and  ir'k,  which  lie  in  the 
plane  of  the  meridian  section  of  the  centered  system  of  spherical  re- 
fracting surfaces;  and,  likewise,  a  collinear  relation  between  the  plane- 
systems  7r^_!  and  TTfc,  which  lie  in  the  planes  of  the  sagittal  sections 
of  the  astigmatic  bundles  of  rays  before  and  after  refraction  at  the 
kth  surface.  In  Art.  76  we  saw  that  the  chief  rays  before  and  after 
refraction  at  this  surface  were  to  be  regarded  as  the  Principal  Axes 
of  each  of  these  two  pairs  of  collinear  plane  systems.  And  since  the 
chief  ray  after  refraction  at  the  kth  surface  is  identical  with  the  chief 
ray  before  refraction  at  the  (k  +  i)th  surface,  the  following  is  the 
state  of  things  which  we  have  here : 

The  Principal  Axis  of  the  Image-Space  of  the  kth  surface  is  at  the 
same  time  the  Principal  Axis  of  the  Object-Space  of  the  (k  +  i)th 
surface;  and  it  will  be  recalled  that  this  is  precisely  the  one  condition 
that  was  assumed  in  Chapter  VII,  Art.  52,  in  deriving  the  formulae 
for  finding  the  determining-constants  of  a  compound  system  due  to 
the  combination  of  any  number  of  given  simpler  systems.  Thus,  if 
we  know  the  positions  of  the  Focal  Points  Jk,  I'k  and  Jk,  Tk  and  the 
magnitudes  of  the  Focal  Lengths /«,*,  eu^  and  /„,£,  eUtk  for  the  Meridian 
and  Sagittal  Rays,  respectively,  for  each  one  of  the  m  spherical  surfaces 
of  the  centered  system,  we  can  employ  straightway  the  formulae  re- 
ferred-to  above,  in  order  to  determine  the  positions  of  the  Focal  Points 
J,  I'  and  /,  T  and  the  magnitudes  of  the  Focal  Lengths  /„,  eu  and  Ju,  eu 
of  the  entire  compound  system. 

Obviously,  we  may  also  employ  here  exactly  the  same  method  as 
was  used  in  Chapter  VIII,  Art.  54,  for  finding  the  Focal  Lengths  of  a 
centered  system  of  spherical  refracting  surfaces  for  the  case  of  Paraxial 
Rays.  Thus,  for  the  Sagittal  Rays  we  should  find  without  difficulty: 


For  the  case  of  the  Meridian  Rays,  since  (Fig.  130) 


BkUk          COSQ!A' 

'we  should  find,  in  the  same  way,  the  following  formula: 

cos  ay  cos  a2  •  •  •  cos  am    yy-  -^.- 

~~    —  —  "~ 


e  .  = 


m 


cos  «L  •  cos  a2  •  •  •  cos  am      S2  •  S3  •  •  •  sm 
Thus,  having  found  by  means  of  formulae  (260)  and  (261)  the  magni- 


360  Geometrical  Optics,  Chapter  XI.  [  §  249. 

tudes  of  the  two  Secondary  Focal  Lengths  e'u  and  e'u,  the  magnitudes  of 
the  Primary  Focal  Lengths  fu  and  fu  can  be  calculated  from  the  fol- 
lowing relations: 

4  -£--4.  (262) 


ART.  78.     SPECIAL  CASES. 

249.  The  Special  Case  of  the  Refraction  of  a  Narrow  Bundle  of 
Rays  at  a  Plane  Surface. 

When  we  are  given  a  chief  ray  u  incident  at  a  certain  point  B  of 
a  spherical  refracting  surface,  we  have  seen  how  we  can  construct  the 
corresponding  refracted  ray  u'  (Chapter  I  X,  §  206)  and  determine  the 
position  of  a  certain  fixed  point  K  (§  234),  which  is  the  centre  of  per- 
spective of  the  range  of  Object-Points  lying  on  u  and  the  corresponding 
range  of  I.  Image-Points  lying  on  u'  ',  just  as  the  centre  C  of  the  sphere 
is  also  the  centre  of  perspective  of  the  range  of  Object-  Points  lying 
on  u  and  the  range  of  II.  Image-Points  lying  on  uf.  We  saw  also 
that  when  the  radius  of  the  spherical  surface  varies,  these  points  C 
and  K  do  not  remain  fixed,  but  move  along  two  fixed  straight  lines.  In 
particular,  if  the  radius  of  the  refracting  surface  becomes  infinite,  so 
that  this  surface  is,  therefore,  a  Plane  Surface,  the  points  C  and  K 
will  be  the  infinitely  distant  points  of  the  two  fixed  straight  lines. 
And,  hence,  in  the  case  of  a  Plane  Refracting  Surface,  as  was  shown 
in  Chapter  III,  Art.  20,  the  straight  lines  joining  the  Object-Points 
lying  on  the  chief  incident  ray  u  with  their  corresponding  II.  Image- 
Points  lying  on  u'  will  all  be  parallel  to  the  fixed  straight  line  BC 
normal  to  the  refracting  plane;  and,  similarly,  the  straight  lines  join- 
ing the  Object-Points  lying  on  u  with  their  corresponding  I.  Image- 
Points  lying  on  u'  will  all  be  parallel  to  the  other  fixed  straight  line  B  K. 
In  this  special  case,  therefore,  the  range  of  Object-Points  on  u  and  the 
two  ranges  of  I.  and  II.  Image-Points  on  u'  are  three  similar  ranges  of 
points.1 

The  refracted  ray  u'  corresponding  to  a  given  ray  u  incident  on 
a  plane  refracting  surface  /i/z  at  the  point  B  (Fig.  131)  may  be  con- 
structed by  using  YOUNG'S  Construction,  as  follows: 

On  the  incidence-normal  take  a  point  0,  and  with  this  point  as 
centre  and  with  radii  equal  to  n'-OB/n  and  n-OBJn'  describe  in 

1  See  F.  LIPPICH:  Ueber  Brechung  und  Reflexion  unendlich  duenner  Strahlensysteme 
an  Kugelflaechen:  Denkschr.  der  kaiserl.  Akad.  der  Wissenschaften  zu  Wien,  xxxviii.  (1878), 
163-192. 

Also,  F.  KESSLER:  Beitraege  zur  graphischen  Dioptrik:  Zft.  J.  Math.  u.  Phys.,  xxix. 
(1884),  65-74. 


§  250.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  361 

the  plane  of  incidence  the  arcs  of  two  concentric  circles  clt  c2.  If  Zl 
designates  the  point  of  intersection  of  the  incident  ray  u  with  the 
arc  clt  and  if  Z2  designates  the  point  of  intersection  of  the  straight 
line  OZl  with  the  arc  c2,  the  straight  line  BZ2  will  be  the  path  of 
the  refracted  ray  u'. 

The  normal  to  the  plane  refracting  surface  gives  the  direction  of 
the  infinitely  distant  point  C.     The  direction  of  the  infinitely  distant 


FIG.  131. 

REFRACTION  OF  INFINITELY  NARROW  BUNDLE  OF  RAYS  AT  A  PLANE  SURFACE.  Construction 
of  Chief  Refracted  Ray  u'  Corresponding  to  Chief  Incident  Ray  u ;  and  Construction  of  I.  and  II. 
Image-Points  S'  and  S'  corresponding  to  a  given  Object  Point  S  on  u.  Centres  of  Perspective  Cand 
K  both  at  infinity.  Plane  Surface  is  regarded  as  a  Spherical  Surface  with  Infinite  Radius. 

point  K  is  found  by  drawing  OY  perpendicular  to  BZl  and  YH  per- 
pendicular to  OZj.  Then  the  point  K  will  be  the  infinitely  distant 
point  of  the  straight  line  B  H. 

The  I.  Image-Point  S'  and  the  II.  Image-Point  3'  corresponding  to 
an  Object- Point  5  on  the  chief  incident  ray  u  are  found  by  drawing 
through  S  straight  lines  parallel  to  BK  and  BC,  which  will  meet  the 
chief  refracted  ray  u'  in  the  required  points  S'  and  3'  respectively. 
The  Focal  Points  of  the  Meridian  and  Sagittal  Rays  coincide  with 
the  infinitely  distant  points  of  the  chief  incident  and  refracted  rays. 
,  By  putting  r  =  oo  in  the  formulae  of  Arts.  71  and  72  of  this  chapter, 
we  shall  derive  immediately  the  same  formulae  as  were  obtained  in 
Chapter  III,  Art.  19. 

250.  Reflexion  at  a  Spherical  Mirror  Treated  as  a  Special  Case 
of  Refraction  at  a  Spherical  Surface. 

In  the  case  of  Reflexion  (n'/n  =  —  i),  we  cannot  use  YOUNG'S 


362 


Geometrical  Optics,  Chapter  XI. 


[  §  250. 


Construction  for  constructing  the  reflected  ray  u'  corresponding  to  a 
ray  u  incident  on  a  spherical  mirror,  for  the  obvious  reason  that  the 
auxiliary  spherical  surfaces  T  and  r',  and  with  them  the  Aplanatic 
Points  Z,  Z',  used  in  this  construction  (§§  206,  207),  have  here  no 
meaning.  Except,  however,  such  properties  as  depend  on  these  par- 
ticular features,  we  have  in  the  case  of  Reflexion  at  a  Spherical  Mirror 
relations  corresponding  precisely  to  those  which  we  found  in  the  in- 
vestigation of  Refraction  at  a  Spherical  Surface.  It  is  very  easy  to 
obtain  these  relations  independently,  but  it  is  also  instructive  to  con- 
sider the  problem  as  a  special  case  of  refraction  (§  26). 

If  in  Fig.  132,  where  C  designates  the  position  of  the  centre  of 
the  Spherical  Mirror  MM,  the  chief  incident  ray  u  meets  the  mirror 

at  the  point  B,  the  corre- 
spond ing  re  fleeted  ray  u'  will 
have  a  direction  such  that 
Z  CBu  =  £u'BC.  On  CB 
as  diameter  describe  a  circle 
cutting  u,  u'  in  the  points  Y, 
Yf,  respectively.  O  b  v  i- 
ously,  exactly  as  was  the 
case  in  refraction,  the  point 
F'  on  u'  is  the  I.  Image- 
Point  of  the  Object-Point  Y 
on  u  (§  233);  and,  hence,  the 
centre  of  perspective  K 
(§  234)  of  the  range  of  Ob- 
ject-Points on  u  and  the 
range  of  corresponding  I. 
Image-Points  on  uf  will  lie 
on  the  straight  line  YY'. 

The  actual  position  of  K  is  found  by  drawing  CK  perpendicular  to 
YY'\  thus,  K  is  seen  to  be  the  point  of  intersection  of  the  straight  lines 
YY'  and  CB. 

The  I.  and  II.  Image-Points  5'  and  3'  on  the  chief  reflected  ray  u' 
corresponding  to  an  Object-Point  5  on  the  chief  incident  ray  u  are 
determined  by  drawing  from  5  straight  lines  through  K  and  C\  the 
intersections  of  5^  and  5  C  with  u'  will  determine  the  points  S'  and  3', 
respectively.  Straight  lines  drawn  through  K  and  C  parallel  to  the 
ray  u  will  determine  by  their  intersections  with  the  reflected  ray  uf 
the  Focal  Points  /'  and  /',  respectively.  Similarly,  the  Focal  Points 
J  and  J  on  u  are  found  by  drawing  through  K  and  C,  respectively, 
straight  lines  parallel  to  u'. 


FIG.  132. 

REFLEXION  OF  INFINITELY  NARROW  BUNDLE  OF 
RAYS  AT  A  SPHERICAL  MIRROR,  u,  u'  Chief  Incident 
and  Reflected  Rays,  respectively. 


=  a  =  Z  S'BC. 


s, 


§251.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface. 


363 


The  Metric  Relations  which  we  have  for  the  case  of  the  Reflexion 
of  a  narrow  bundle  of  rays  at  a  Spherical  Mirror  can  be  derived  from 
the  corresponding  Refraction-Formulae,  which  have  been  obtained  in 
this  chapter,  by  merely  putting  n'  =  —  n  and  a'  =  —  a.  However, 
in  the  formulae  derived  in  this  way,  the  reader  should  bear  in  mind, 
that,  according  to  the  convention  we  made  in  §  26,  the  positive  direction 
of  any  straight  line  is  the  direction  along  that  line  which  light  would 
pursue  if  the  line  were  the  path  of  an  incident  ray,  and,  accordingly, 
the  positive  direction  along  a  reflected  ray  is  the  direction  exactly 
opposite  to  that  in  which  the  reflected  light  is  propagated  along  it. 
Failure  to  note  this  point  has  been  a  source  of  frequent  confusion 
with  writers  on  Optics. 

We  derive,  therefore,  the  following  set  of  Formula  for  the  Reflexion 
of  a  Narrow  Bundle  of  Rays  at  a  Spherical  Mirror: 


I.  Meridian  Rays: 


CK  =  ;"sin2a;,      Z.BKC  =  o; 

T7_       _,  _,  r-cosa 

=  e,,  =  JB  =  I  B  =  —  - 


i  +  i— 

5      ^       r • cos  a 


«•-?• 


II.  5<zgi«o/  Rays: 


2  cos  a 


;  2, 


2  cos  a 

s 

=  -, . 
5 


(263) 


251.     Astigmatism  of  an  Infinitely  Thin  Lens. 

Provided  we  assume  that  the  length  of  the  path  of  the  chief  ray 
within  the  Lens  is  negligible  (which  may  sometimes  be  a  rather  big 
assumption,  even  though  the  Lens  is  infinitely  thin),  and  accordingly 
put  B1B2  =  o,  we  shall  have: 


and  since  here  there  is  no  possibility  of  confusion,  we  shall  find  it 
convenient  to  write:  5  =  slf  s'  =  s'2  and  s  =  slt  s'  =  s2.  Moreover, 
since  the  Lens  is  supposed  to  be  surrounded  by  the  same  medium  on 
both  sides,  we  may  also  write:  «t  =  n'2  =  n,  n\  =  n'.  Thus,  for  the 
case  of  an  Infinitely  Thin  Lens  (m  =  2),  formulae  (259)  give  the  fol- 


364  Geometrical  Optics,  Chapter  XI. 

lowing  relations: 

v  _  n'  -  cos  <x(  —  n  -  cos  ^  n-  cos  a'2  —  n'  •  cos  a2 

*\  ~   ~  ~  ~~  i         *2  ~  ~  ~  ~"  i 

ri  r2 

I.  Meridian  Rays : 

L  _  cos2  <y  cos2  a2    I         cos2  «2    /     Vl  V2     \ 

s'      cos2  a'i  -  cos2  a'2    s       n-  cos2  a'2  \  cos2  o^      cos2  «2 /  ' 


[§251. 


II.  Sagittal  Rays: 


(264) 


The  conditions  that  to  an  Object-Point  S  lying  on  the  chief  object- 
ray  u  there  shall  correspond  a  "  Homocentric"  Image-Point  S'  lying  on 
the  chief  image-ray  u'  are  s  =  s  =  B2,  sr  =  s'  =  B2',  whence  we  find : 


n(cos2  o?!  •  cos2  a2  —  cos2  a\  -  cos2  «2) 

a(  -cos2  «2  —  cos2  a2)  —  F2-cos2  &[  -sin2  a^ 


•     (265) 


In  general,  therefore,  on  every  incident  chief  ray  u  there  is  one  such 
Object-Point  S  to  which  on  the  corresponding  emergent  chief  ray  uf 
there  corresponds  a  "Homocentric"  Image-Point  S'. 

A  case  of  both  theoretical  and  practical  interest  occurs  when  the 
chief  ray  goes  through  the  Optical  Centre  of  the  Infinitely  Thin  Lens 
(which  is  easily  contrived  by  placing  a  screen  with  a  small  circular 
opening  right  in  front  of  the  Lens).  In  this  case  the  paths  of  the 
incident  and  emergent  chief  rays  are  along  the  same  straight  line,  and, 
accordingly,  we  have: 

a\  =  a2>     and  also     al  =  o:2; 
and,  therefore, 

Introducing  these  values  in  formulae  (264)  above,  we  obtain  for  this 
special  case: 

Formula  for  Calculating  the  Astigmatism  of  an  Infinitely  Thin  Lens 
for  the  case  when  the  Chief  Ray  goes  through  the  Optical  Centre  : 

I.  Meridian  Rays : 


n 


II.  Sagittal  Rays : 


i       i 

?~J 


F1/2- 
n        r» 


,   ^x 
(266) 


§  251.]       Refraction  of  Narrow  Bundle  of  Rays  at  Spherical  Surface.  365 

The  positions  of  the  Secondary  Focal  Points  /'  and  I'  of  the  systems 
of  Meridian  and  Sagittal  Rays  of  the  astigmatic  bundle  of  emergent 
rays  may  be  found  by  putting  5  =  s  =  °o  in  the  above  formulae. 
Thus,  if  A  designates  the  position  on  the  axis  of  the  Optical  Axis  of 
the  Thin  Lens,  we  have 


-  r,) 


(26) 


If  in  formulae  (261)  and  (262),  we  introduce  the  special  conditions 
which  we  have  in  the  present  case,  viz.  :  m  =  2,  s{  =  s2,  a{  =  a'2,  a{  =  a2 
and  WL  =  n'2,  we  find  for  the  Focal  Lengths  of  the  system  of  Meridian 
Rays: 

/.--«:-  Ai'. 

In  the  same  way,  formulae  (260)  and  (262),  give  for  the  Focal  Lengths 
of  the  system  of  Sagittal  Rays: 

7.  -  -  r.  -  AT.  •         - 

Accordingly,  in  the  special  case  which  we  have  here  the  Focal  Lengths 
of  both  systems  of  rays  are  equal  to  the  distances  of  the  Focal  Points 
from  the  incidence-point  A.  Thus,  we  have: 

-cos2  «!  =    T7--^~    -cos2^;    (268) 


so  that  now  formulae  (266)  may  be  put  in  the  following  forms: 

I.  Meridian  Rays  :       ijs'  —  i/s  =  i//u;  "] 

r  (269) 

II.  Sagittal  Rays:        i/sf  -  i/s  =  i/fu.  } 

These  equations,  as  will  be  immediately  recognized,  have  precisely  the 
same  form  as  the  formula  for  the  Refraction  of  Paraxial  Rays  through 
an  Infinitely  Thin  Lens,  formula  (99)  of  Chap.  VI.  The  Focal  Lengths 
/„  and  /M  are  both  functions  of  the  slope-angle  «L  of  the  chief  ray,  and 
for  the  value  at  =  o  we  obtain  : 

(«i  =  o),    /„=?„=/  =  nr^Kn'  -  n)(r2  -  r,). 

When  the  chief  ray  goes  through  the  Optical  Centre  of  the  Infinitely 
Thin  Lens,  the  Astigmatic  Difference  vanishes,  in  general,  only  for 


366  Geometrical  Optics,  Chapter  XI.  [  §  251. 

the  case  when  the  Object-Point  is  in  contact  with  the  Lens;  but  for 
<*!  =  o  it  vanishes  for  all  positions  of  the  Object-Point.1 

Another  interesting  special  case  which  has  been  investigated  by  H. 
HARTING2  is  the  case  when  the  chief  ray  crosses  the  optical  axis  at 
the  common  vertex  of  a  System  of  Thin  Lenses  in  Contact. 

1  See  Die  Theorie  der  optischen  Instrumente,  edited  by  M.  VON  ROHR  (Bd.  I,  Berlin, 
1904);  IV.  Kapitel,  "  Die  Realisierung  der  optischen  Abbildung",  von  P.  CULMANN,  S.  179. 

2  H.  HARTING:  Einige  Bemerkungen  zu  dem  Aufsatze  des  Hrn.  B.  WANACH:  Ueber  L. 
v.  SEIDELS  Formeln  zur  Durchrechnung  von  Strahlen  u.  s.  w.:   Zft.  f.  Instr.,  xx.  (1900), 
234-237.     See  also  Die  Theorie  der  optischen  Instrumente,  edited  by  M.  VON  ROHR  (Bd.  I, 
Berlin,  1904);  V.  Kapitel,  "  Die  Theorie  der  sphaerischen  Aberrationen",  S.  254. 


CHAPTER    XII. 

THE   THEORY   OF   SPHERICAL   ABERRATIONS. 

I.     INTRODUCTION. 
ART.  79.     PRACTICAL  IMAGES. 

252.  The  requirements  of  a  good  image  are  (i)  that  it  shall  be 
sharp  or  distinct,  corresponding,  therefore,  to  the  object  point  by  point, 
(2)  that  it  shall  be  accurate,  that  is,  completely  similar  to  the  object, 
and  thus  faithfully  reproducing  it,  and  (3)  that  it  shall  be  bright. 
This  last  condition  necessarily  implies  the  use  of  wide-angle  bundles 
of  rays,  because  obviously  the  light-intensity  at  any  point  will  be 
greater  in  proportion  to  the  number  of  rays  that  unite  at  that  point. 
On  the  other  hand,  the  first  two  requirements,  which  are  both  purely 
geometrical,  will,  in  general,  be  fulfilled  by  an  optical  system  only  in 
the  special  and  unrealizable  case  when  the  bundles  of  rays  concerned 
in  the  production  of  the  image  are  infinitely  narrow.  Thus,  in  the 
theory  of  the  Imagery  by  means  of  Paraxial  Rays,  which  was  developed 
according  to  general  laws  first  by  GAUSS,  and  which  has,  therefore, 
been  appropriately  called  "GAiissian  Imagery"  (§  188),  we  have  seen 
that  for  an  optical  system  of  centered  refracting  (or  reflecting)  spher- 
ical surfaces  a  distinct  and  accurate  image  was  formed  only  when  the 
rays  concerned  were  all  comprised  within  an  indefinitely  narrow  cylin- 
drical space  immediately  surrounding  the  optical  axis;  this  region  be- 
ing more  explicitly  defined  by  the  condition  that  a  "paraxial"  ray  is  one 
for  which  both  the  angle  of  incidence  a  and  the  central  angle  <p  were 
so  small  that  all  powers  of  these  angles  higher  than  the  first  could  be 
neglected  (§  109). 

In  general,  even  with  infinitely  narrow  bundles  of  rays,  stigmatic 
imagery,  except  in  the  case  of  normally  incident  rays  just  mentioned, 
is  possible  only  for  certain  special  positions  of  the  object-point. 

It  goes  without  saying  that  from  the  standpoint  of  the  optician  the 
formation  of  images  under  such  impracticable  restrictions  is  almost 
without  interest.  Without  dwelling  on  the  obvious  objection  that 
such  images  would  be  of  infinitesimal  dimensions  (as  would  be  like- 
wise true  of  the  objects  to  be  depicted),  we  encounter  a  still  greater 
difficulty  in  the  fact  that  Physical  Optics — which  in  all  optical  ques- 
tions is  the  court  of  last  resort — pronounces  that  these  images  are 
not  true  images  at  all.  For  according  to  the  Wave-Theory  of  Light, 

367 


368  Geometrical  Optics,  Chapter  XII.  [  §  252. 

a  mere  homocentric  convergence  of  the  image-rays  is  not  of  itself  suf- 
ficient for  the  formation  of  a  distinct  optical  image.  If  the  wave-surf* 
ace  in  the  Image-Space  is  spherical — so  that  the  image-rays  all  meet 
in  one  point,  viz.,  at  the  centre  of  the  spherical  wave-surface — instead 
of  an  image-point,  we  shall  obtain  a  resultant  effect  (in  the  plane 
perpendicular  to  the  optical  axis  through  the  centre  of  the  spherical 
wave-surface)  consisting  of  a  central  luminous  disc  surrounded  by 
alternate  dark  and  diminishingly  bright  rings.  The  brightness  of  this 
disc  fades  from  the  centre  out  towards  the  circumference.  The  greater 
the  extent  of  the  effective  portion  of  the  spherical  wave-surface  as 
compared  with  its  radius,  that  is,  the  wider  the  angle  of  the  homo- 
centric  bundle  of  image-rays,  the  smaller  will  be  the  diameter  of  the 
diffraction-disc,  which  therefore  tends  to  be  reduced  more  and  more 
nearly  to  a  point  as  the  angular  aperture  of  the  bundle  of  image-rays 
is  increased.  From  the  point  of  view  of  Physical  Optics,  as  LUMMERX 
remarks,  this  is  the  only  sense  in  which  the  term  "point-image"  can 
have  any  meaning.  Thus,  both  for  a  clear  and  distinct  image  as  well 
as  for  a  bright  image,  theory  insists  that  wide-angle  bundles  of  rays 
must  be  employed  (see  §45). 

On  the  other  hand,  from  the  geometrical  standpoint  the  fundamental 
requirement  of  optical  imagery  is  the  convergence  of  the  rays  to  one 
point;  and,  in  general,  this  requirement  in  the  case  of  bundles  of  finite 
aperture  is  impossible. 

Consequently,  actual  optical  images,  which  are  necessarily  formed 
by  bundles  of  rays  of  finite  aperture,  are,  in  general,  more  or  less 
faulty.  These  faults — which  are  called  aberrations — may  sometimes 
escape  unnoticed  merely  because  the  eye  which  views  the  image  cannot 
or  does  not  distinguish  the  defects  which  it  contains.  But  to  the 
practical  optician  who  strives  to  obtain  an  image  as  nearly  perfect  as 
possible  it  is  of  the  highest  importance  to  ascertain  the  nature  of  these 
various  so-called  aberrations,  to  distinguish  them  the  one  from  the 
other,  and  to  perceive  clearly  what  factors  contribute  to  produce  them 
in  each  instance,  so  that  in  the  design  of  an  optical  instrument  he  may 
contrive  to  reduce,  perhaps  to  abolish  entirely,  at  any  rate  those 
aberrations  which  for  the  particular  type  of  instrument  are  to  be  re- 
garded as  the  most  objectionable.  Along  these  lines,  and  especially 
since  the  rise  of  Photography,  wonderful  progress  has  been  achieved 
in  the  design  and  construction  of  optical  instruments. 

The  plan  that  is  employed  is  to  combine  optical  systems  in  such  a 
way  that,  although  each  single  refracting  or  reflecting  surface  gives 

1  See  MUELLER-POUILLET'S  Lehrbuch  der  Physik  (neunte  Auflage),  Bd.  II,  447. 


§  253.]  Theory  of  Spherical  Aberrations.  369 

by  itself  a  point-to-point  imagery  only  within  the  narrow  region  to 
which  the  paraxial  rays  are  confined,  in  the  compound  system  these 
limitations  are  very  considerably  extended  in  one  direction  or  another 
or  perhaps  in  several  directions  simultaneously.  The  duty  of  refract- 
ing the  rays  so  that  they  will  emerge  finally  in  suitable  directions  is 
not  assigned  to  a  single  surface,  but  is  distributed  over  a  number  of 
separate  surfaces.  By  suitable  combinations,  it  has  been  found  pos- 
sible in  this  way  to  construct  systems  which  by  means  of  wide-angle 
bundles  of  rays  will  give  a  true  image  of  an  axial  object-point  or  of 
a  small  surface-element  placed  at  right  angles  to  the  optical  axis. 
The  objective  of  a  microscope,  for  example,  is  a  system  of  this  kind. 
In  the  eye-piece,  or  ocular,  on  the  other  hand,  we  have  an  illustration 
of  a  system  which  by  means  of  relatively  narrow  bundles  of  rays  pro- 
duces the  image  of  a  large  object.  Thus,  in  the  compound  microscope 
the  duty  of  the  objective  is  to  produce  an  image  of  a  small  object  by 
means  of  wide-angle  bundles  of  rays,  whereas  the  duty  of  the  ocular 
is,  by  means  of  narrow  bundles,  to  spread  over  the  large  field  of  vision 
the  image  produced  by  the  objective.  In  the  case  of  the  photographic 
objective,  we  must  have  both  wide  aperture  and  extensive  field  of 
vision,  and  in  order  to  meet  both  of  these  requirements  at  once,  some- 
thing else  has  to  be  sacrificed,  and,  accordingly,  we  are  obliged  to  be 
content  with  a  less  distinct  image  than  we  require  in  the  case  of  the 
objective  of  a  microscope. 

Of  course,  it  would  be  idle  for  the  optician  to  seek  to  produce  an 
image  which  is  free  from  faults  that  could  not  be  detected  by  the  eye 
if  they  were  present.  The  resolving-power  of  the  human  eye  is  com- 
paratively poor  (cf.  §  377).  Thus,  for  example,  details  in  the  object 
which  are  separated  by  an  angular  distance,  say,  of  one  minute  will  riot 
be  recognized  by  the  eye  as  separate  and  distinct.  Accordingly,  the 
practical  image  need  be  perfect  only  to  the  degree  that  in  it  those 
elements  of  the  object  which  are  to  be  preceived  as  separate  must  be 
presented  to  the  eye  at  a  visual  angle  of  not  less  than  one  minute  of  arc. 

ART.  80.     THE  SO-CALLED  SEIDEL  IMAGERY. 

253.  The  theory  developed  by  GAUSS1  in  his  Dioptrischen  Unter- 
suchungen  proceeds  on  the  assumption  that  the  central  angle  <p  is  so 
small  that  the  second  and  higher  powers  thereof  are  negligible.  The 
theory  is  applicable,  therefore,  only  to  optical  systems  of  narrow  aper- 
ture and  of  small  visual  field,  since  both  the  incidence-points  of  the 
rays  and  the  object-points  whence  they  emanate  must  all  lie  very  close 

1  C.  F.  GAUSS:  Dioptrische  Untersuchungen  (Goettingen,  1841). 

25 


370  Geometrical  Optics,  Chapter  XII.  [  §  254. 

to  the  optical  axis  of  the  centered  system  of  spherical  surfaces.  The 
investigations  of  EULER/  ScHLEiERMACHER,2  SEIDEL3  and  others 
were  first  directed  towards  taking  account  of  the  aberrations  due  to 
increase  of  the  aperture  of  the  system;  but,  later,  with  the  rise  of 
Photography  and  the  development  of  the  Photographic  Objective, 
it  became  necessary  to  take  into  consideration  not  only  a  wider  aper- 
ture but  a  greater  field  of  vision,  in  order  to  portray  objects  which 
were  at  some  distance  from  the  optical  axis.  The  complete  theory  of 
spherical  aberrations  was  worked  out  by  J.  PETZVAL4  and  L.  SEIDEL ,5 
and  in  the  following  sections  of  this  article  the  methods  of  these  two 
investigators  form  the  basis  of  the  mode  of  treatment. 

254.  Order  of  the  Image,  according  to  J.  Petzval.  Taking  the 
optical  axis  of  the  centered  system  of  spherical  surfaces  as  the  #-axis 
of  a  system  of  rectangular  co-ordinates,  let  us  denote  the  co-ordinates 
of  an  object-point  P  by  £,  77,  f .  The  transversal  plane  a,  which  passes 
through  P  and  is  perpendicular  to  the  optical  axis,  will  be  called  the 
Object- Plane.  Let  P  (|,  T|,  £)  designate  the  position  of  the  point  where 
the  rectilinear  path  of  an  object-ray,  proceeding  from  the  object-point 
P,  crosses  a  second  fixed  transversal  plane  cr  parallel  to  the  object- 
plane  <7.  The  position  of  this  object-ray  will  be  completely  determined 
by  the  four  parameters  77,  f ,  i\,  £.  In  the  image-space  let  cr',  <r'  designate 
a  pair  of  fixed  transversal  planes  perpendicular  to  the  optical  axis;  we 
shall  call  the  plane  a'  the  Image-Plane.  Let  P',  P'  designate  the  posi- 
tions of  the  points  where  the  rectilinear  path  of  the  image-ray,  corre- 
sponding to  the  object-ray  PP,  crosses  the  planes  cr',  or',  respectively, 
and  let  the  rectangular  co-ordinates  of  P',  P'  be  denoted  by  (£',  77',  £ ') 
and  by  (§',  i]',  £'),  respectively.  The  position  of  the  image-ray  will, 
therefore,  be  defined  by  the  four  parameters  r;',  £',  TJ',  £'. 

Since  to  every  object-ray  there  corresponds  one,  and  only  one, 

1L.  EULER:  Dioptricce  pars  prima  (Petersburg,  Akad.  Wiss.,  1769);  pars  secunda 
(ibid.,  1770);  pars  tertia  (ibid.,  1771). 

2  L.  SCHLEIERMACHER  :  Ueber  den  Gebrauch  der  analytischen  Optik  bei  Construction 
optischer  Werkzeuge  (Pocc.  Ann.,  1828,  xiv.);  also,  Analytische  Optik  (BAUMGARTNERS 
und  VON  ETTINGSHAUSENS  Zft.  f.  Phys.  u.  Math.,  1831,  ix.,  1-35;  161-178;  454-474: 
1832,  x.,  171-200;  329-357). 

3L.  SEIDEL:  Zur  Theorie  der  Fernrohr  Objective:  Astr.  Nachr.,  1853,  xxxv.  No.  836, 
301-316. 

4  JOSEPH  PETZVAL  :  Bericht  ueber  die  Ergebnisse  einiger  dioptrischer   Untersuchungen 
(Pesth,  1843).     See  also:  Bericht  ueber  optische  Untersuchungen.    Sitzungsber.  der  math.- 
naturwiss.  Cl.  der  kaiserl.  Akad.  der  Wissenschaften,  Wien,  xxvi.  (1857),  50-75.  92-105, 
129-145.     (See  especially  page  95,  in  regard  to  the  "  order  "  of  the  image.) 

5  L.   SEIDEL  :    Zur  Dioptrik.      Ueber  die   Entwicklung    der    Glieder   3ter  Ordnung, 
welche  den  Weg  eines  ausserhalb  der  Ebene  der  Axe  gelegenen  Lichtstrahles  durch  ein 
System  brechender  Medien  bestimmen:  Astr.    Nachr.,  1856,  xliii.,  No.   1027,  289-304; 
No.  1028,  305-320;  No.  1029,  321-332. 


§  255.]  Theory  of  Spherical  Aberrations.  371 

image-ray,  it  is  obvious  that  each  of  the  four  parameters  of  the  image- 
ray  must  be  a  definite  function  of  the  four  parameters  of  the  object- 
ray,  so  that  we  may  write  : 


=/id7,  r,  n,  5), 
,  i.  £)» 


where  the  f  unctions  /!,/2,/3,/4  can  be  deduced  by  the  laws  of  refraction. 

Moreover,  taking  account  of  the  symmetry  with  respect  to  the 
optical  axis,  we  observe  that  if  the  signs  of  the  parameters  77,  f  ,  TJ,  £ 
are  all  reversed,  the  signs  of  the  parameters  77',  f',  T\',  £'  will  all  like- 
wise be  reversed;  and,  consequently,  if  each  of  the  functions  above  is 
developed  in  a  series  of  ascending  powers  and  products  of  77,  f,  TJ,  5, 
each  of  these  series  can  contain  only  the  terms  of  the  odd  degrees. 
And,  hence,  if  the  parameters  of  the  ray  are  regarded  as  magnitudes 
of  the  first  order  of  smallness,  these  series-developments  will  contain 
only  terms  of  the  odd  orders  of  smallness. 

Now,  if  for  all  rays  proceeding  from  the  object-point  P  we  obtain 
exactly  the  same  values  of  the  co-ordinates  77',  £',  we  shall  obtain  at 
P'  a  perfect  image  of  the  object-point  P.  In  general,  however,  this 
will  not  be  the  case,  and  for  a  second  object-ray  coming  from  P,  whose 
parameters  are,  say,  77,  £,  t\  +  8t\,  £  +  5£,  we  shall  obtain  a  new  set 
of  values  77'  +  Si/,  f  +  #',  V  +  «V»  5'  +  5£'  for  a11  four  of  the  par- 
ameters of  the  corresponding  image-ray.  Obviously,  in  the  series- 
developments  the  differences  Sr;',  6£'  will  contain  also  only  the  terms 
of  the  odd  degrees.  If,  as  compared  with  the  magnitudes  77,  f,  i\,  £, 
these  differences  £77',  d£'  are,  say,  of  the  (2k  +  Oth  order  of  smallness, 
then,  according  to  J.  PETZVAL,  the  spot  of  light  formed  around  P' 
by  the  totality  of  all  such  points  as  P'  is  to  be  considered  as  an  "image" 
of  the  (2k  +  i)th  order  in  the  image-plane  <r'  corresponding  to  the 
object-point  P.  The  higher  the  order  of  the  image,  the  more  nearly 
perfect  it  will  be.  An  image  of  the  3rd  order  is  one  in  which  there 
are  uncorrected  faults  of  the  3rd  order. 

255.  Parameters  of  Object-Ray  and  Image-Ray,  according  to  L. 
Seidel.  A  complete  development  of  the  theory  of  Spherical  Aberra- 
tions was  first  published  by  L.  SEIDEL,  who  extended  GAUSS'S  theory 
so  as  to  take  account  of  magnitudes  of  the  3rd  order  of  smallness, 
neglecting  therefore  the  terms  of  the  5th  and  higher  orders.  Thus, 
in  the  so-called  SEIDEL  Imagery,  the  image  is  of  the  fifth  order. 

The  comparative  simplicity  and  elegance  of  SEIDEL'S  methods  are 
due  to  his  choice  of  the  four  parameters  which  define  the  rectilinear 
path  of  the  ray,  viz.,  the  two  pairs  of  rectangular  co-ordinates  (77,  f) 


372  Geometrical  Optics,  Chapter  XII.  [  §  255. 

and  (TJ,  £)  of  the  points  P,  P  where  the  ray  crosses  the  two  fixed  trans- 
versal planes  <7,  <r.  In  order  to  make  this  clear,  let  us  suppose  now 
that  PP  represents  the  path,  not  of  the  object-ray  itself,  as  formerly, 
but  of  this  ray  before  refraction  at,  say,  the  kth  surface  of  the  optical 
system,  and,  in  the  same  way,  let  P'Pr  represent  the  path  of  the  ray 
after  refraction  at  this  surface.  The  actual  locations  of  the  four  trans- 
versal planes  or,  a'  and  <r,  or'  have  not  been  specified;  and,  accordingly, 
we  may  establish  an  arbitrary  connection  between,  say,  a  and  tr',  on 
the  one  hand,  and  between  a  and  <r',  on  the  other  hand.  If,  for 
example,  M,  M'  designate  the  points  where  the  optical  axis  meets  the 
planes  cr,  a',  respectively,  these  points  may  be  selected  with  reference 
to  each  other  so  that,  in  the  sense  of  GAUSS'S  Theory,  M,  M'  are  a  pair 
of  conjugate  axial  points  with  respect  to  the  spherical  refracting  sur- 
face which  is  here  under  consideration.  And  the  same  relation  can 
be  established  between  the  pair  of  points  Af ,  M'  where  the  optical 
axis  crosses  the  transversal  planes  or,  <r',  respectively.  Thus,  by 
GAUSS'S  Theory,  the  transversal  planes  cr,  cr'  and  or,  <?'  will  be  two  pairs 
of  conjugate  planes  with  respect  to  the  spherical  surface  in  question. 
If  A  designates  the  vertex  and  C  the  centre  of  this  spherical  surface, 
and  if  we  put: 

AC  =  r,     AM  =  u,     AM'  =  u',     AM  =  u,     AM'  =  u', 

the  relations  between  M  and  M'  and  between  M  and  M'  will  be 
expressed  as  follows  (see  §  1 26) : 


(270) 


(i       i\        ,(i        i\       . 

n[ -  I  =  »  1  -  —  —  I  =  •/. 

\r      u)          \r      u'/ 


The  co-ordinates  of  the  four  points  P,  P,  P',  P'  may  now  be  ex- 
pressed as  follows: 


wherein  the  first  term  on  the  right-hand  side  of  each  of  these  equations 
denotes  the  approximate  (or  "GAUSsian")  value  of  the  parameter  ob- 
tained by  neglecting  the  terms  of  the  3rd  order,  and  the  second  term 
denotes  the  correction  of  the  ^rd  order,  which,  being  added  to  the  prin- 
cipal, or  approximate,  value,  gives  a  value  which  will  be  exact  except 


§  256.]  Theory  of  Spherical  Aberrations.  373 

for  residual  errors  of  the  5th  and  higher  orders.  Evidently,  the  points 
Q(y,  z)  and  £'(/,  2'),  lying  in  the  planes  <r,  a'  and  not  far  from  the 
points  P,  P',  respectively,  are  a  pair  of  conjugate  points  according 
to  GAUSS'S  Theory;  and  the  same  thing  is  true  also  of  the  pair  of 
points  Q(y,  z)  and  Q'(y',  *'),  which  lie  in  the  transversal  planes  <r,  <r' 
and  not  far  from  the  points  P(r\,  £),  P'Cn'»  £'),  respectively. 

256.  The  Correction-Terms  or  Aberrations  of  the  3rd  Order. 
Thus,  SEIDEL  employs  two  independent  systems  of  transversal  planes 
perpendicular  to  the  optical  axis  of  the  centered  system  of  spherical 
surfaces,  so  that  for  each  medium  traversed  by  the  ray  there  is  one 
plane  of  each  system.  The  position  of  the  object-ray  before  refrac- 
tion at  the  first  spherical  surface  is  given  by  assigning  the  co-ordinates 
(fn  fi)»  Oluji)  °f  the  points  Px,  Pl  where  the  ray  crosses  two  arbitrary 
transversal  planes  «rlf  ov 

For  the  plane  <rt  naturally  we  shall  select  the  transversal  plane 
which  contains  the  object-point  P^,  Ti);  this  is  the  so-called  Object- 
Plane  mentioned  above  (§  254).  Moreover,  without  affecting  at  all 
the  generality  of  the  discussion,  we  may  select  the  xy-  plane  of  the 
system  of  rectangular  co-ordinates  so  that  the  object-point  Pl  lies  in 
this  plane,  in  which  case  we  shall  have  £x  =  o.  Since  the  bundle  of 
object-rays  is  homocentric,  the  point  Qi(ylt  %)  will  coincide  with  PL; 
that  is,  5^  =  o,  5^  =  o. 

The  plane  a'k  is  the  transversal  plane,  which,  according  to  GAUSS'S 
theory,  is  conjugate,  with  respect  to  the  first  k  spherical  surfaces  of 
the  optical  system,  to  the  Object-Plane  o^.  After  refraction  at  the 
kih  surface,  the  ray  (prolonged  either  forwards  or  backwards,  if  neces- 
sary) will  cross  the  plane  <r'k  at  the  point  Pk(nkt  fl).  K  m  denotes  the 
total  number  of  spherical  surfaces,  the  corresponding  image-ray, 
emerging  from  the  optical  system,  will  cross  the  Image-Plane  a'm  at 
the  point  P'm,  whose  co-ordinates  are: 


where  y'm,  zm  denote  the  co-ordinates  of  the  point  Q'm  which,  by  GAUSS'S 
theory,  is  the  image  of  the  object-point  Pl  (or  Q^.  The  magnitudes 
y'm,  zm  can  be  determined  by  the  approximate  formulae  of  GAUSS. 
Obviously,  the  point  Q'm  will  lie  in  the  meridian  plane  through  the 
point  <2i,  and  since  Ql  is  coincident  with  Plt  if  the  meridian  plane 
containing  the  object-point  is  taken  as  the  :ry-plane,  we  must  have: 

£1  =  *i  =  zl  =  o; 
and,  hence,  £  =  8z'm. 


374  Geometrical  Optics,  Chapter  XII.  [  §  257. 

The  magnitudes  denoted  by  8y'm,  dzm  are  the  correction-terms,  or 
aberrations  of  the  $rd  order,  which  measure  the  errors  of  the  image. 
By  some  writers  8ym,  dz'm  are  called  the  "Tangential"  and  "Sagittal" 
aberrations,  respectively,  in  the  GAUSsian  Image-Plane  <r'm.  We  may 
also  call  them  the  y-aberration  and  the  z-aberration  in  this  plane. 

257.  Planes  of  the  Pupils  of  the  Optical  System.  So  far  as  the 
meanings  of  the  magnitudes  5y'mJ  8zm  are  concerned,  it  is  a  matter  of 
no  consequence  what  plane  o^  is  selected  for  the  initial  plane  of  the 
other  system  (or  cr-system)  of  transversal  planes.  In  all  optical  in- 
struments the  aperture  of  the  cone  of  effective  rays  is  limited  by  cer- 
tain diaphragms  or  circular  openings,  called  "stops",  which  are  placed 
with  their  planes  perpendicular  to  the  optical  axis  and  with  their  cen- 
tres on  this  axis.  Even  in  case  there  is  no  such  artificial  diaphragm, 
the  rims  or  fastenings  of  the  lenses  themselves  will  act  as  such,  so 
that  of  all  the  rays  emitted  from  an  object-point  only  a  certain  limited 
number  succeed  in  making  their  way  through  the  entire  apparatus. 
When  there  are  several  diaphragms,  the  effective  stop  is  that  one 
which  permits  the  fewest  rays  to  pass.  This  stop  may  be  situated,  ac- 
cording to  circumstances,  in  front  of  the  entire  system  or  somewhere 
within  the  system  or  even  beyond  the  entire  system.  It  is  found  to  be 
most  convenient  to  select  the  plane  o^  so  that  one  of  the  planes  of  this 
series  of  transversal  planes  shall  coincide  with  the  plane  of  the  effect- 
ive stop.  In  the  most  general  case,  when  the  stop  is  situated  within 
the  optical  system,  say,  between  the  &th  and  the  (k  +  i)th  spherical 
surfaces,  the  plane  of  the  stop  will  be  the  transversal  plane  o^,  and  the 
stop-centre  will  be  at  the  point  M'k  where  the  optical  axis  crosses  this 
plane.  The  axial  point  Mlt  whose  image  produced  by  the  refraction  of 
paraxial  rays  through  the  first  k  spherical  surfaces  of  the  optical  system 
is  M^,  will  determine,  therefore,  the  position  of  the  initial  transversal 
plane  ov  All  the  object-rays  cross  the  plane  o^  at  points  lying  within 
the  space  which  would  be  covered  by  a  thin  circular  disc  placed  with 
its  centre  on  the  optical  axis  at  Ml  and  perpendicular  to  the  optical 
axis  and  of  such  dimensions  that,  with  respect  to  the  first  k  surfaces 
of  the  optical  system,  its  GAUSsian  image  at  M'k  exactly  coincided 
with  the  effective  stop  there.  This  circle  around  Ml  in  the  plane  <TI 
has  been  well  called  by  ABBE  the  Entrance- Pupil  (see  §  361)  of  the 
optical  system;  and  we  shall,  therefore,  speak  of  the  initial  plane  <rl  as 
the  "Plane  of  the  Entrance-Pupil".  Similarly,  all  the  image-rays  will 
cross  the  last  transversal  plane  cr'm  in  points  contained  within  a  circle 
around  Afm1  which,  with  respect  to  the  entire  system,  is  the  image, 
by  GAUSS'S  theory,  of  the  Entrance-Pupil.  This  circle  is  called  the 


§  259.]  Theory  of  Spherical  Aberrations.  375 

Exit-  Pupil,  and  the  plane  <r'm  is  called  the  "Plane  of  the  Exit-Pupil". 

Obviously,  by  this  method  of  selecting  the  initial  plane  o^  we  have 
the  advantage  of  knowing  the  greatest  possible  values  which  the  co- 
ordinates y±,  zl  can  have  in  the  case  of  a  given  optical  system,  and, 
since  the  values  of  5y'm,  dzm  depend  also  on  the  values  of  ylt  zlt  this 
knowledge  will  be  of  service  in  considering  the  relative  importance  of 
the  various  terms  in  the  series-developments. 

258.  Chief  Ray  of  Bundle.  Of  all  the  rays  proceeding  from  the 
object-point  Pl  there  is  one,  which,  lying  in  the  meridian  plane  through 
Plt  will,  in  traversing  the  medium  in  which  the  stop  is  situated,  go 
through  the  centre  of  the  stop.  This  ray,  distinguished  as  the  chief 
ray  of  the  bundle,  will,  in  general,  cross  the  optical  axis  for  the  first 
time  at  a  point  Lt  not  very  far  from  the  centre  Ml  of  the  Entrance- 
Pupil.  The  slope  of  the  chief  ray  of  the  bundle  of  object-rays  emanat- 
ing from  Pv  is 


more  exactly  defined  by  the  following  equation  : 

--'  (272) 


where  vl  =  A^  denotes  the  abscissa,  with  respect  to  the  vertex  Al 
of  the  first  surface,  of  the  point  Lx.  Of  course,  under  certain  circum- 
stances the  point  L^  may  coincide  with  Mlt  as  is  often  the  case. 

259.  Relative  Importance  of  the  Terms  of  the  Series-  Develop- 
ments of  the  Aberrations  of  the  3rd  Order.  The  maximum  value 
of  yl  will  be  fixed  by  the  limits  of  the  required  field  of  vision,  and  in 
the  same  way  the  maximum  values  of  ylt  zl  will  depend  on  the  size 
of  the  aperture  of  the  optical  system.  Thus,  for  example,  in  the  case 
of  an  optical  system  of  relatively  small  field  of  vision,  and,  on  the 
other  hand,  of  relatively  large  aperture,  the  most  important  terms  in 
the  series-developments  of  the  aberrations  8y'm,  8z'm  will  be  the  terms 
which  do  not  contain  yl  at  all,  that  is,  the  terms  y\,  y\zlt  y^z\  and 
z\.  Next  in  importance  will  be  the  terms  which  contain  the  first 
power  of  ylt  viz.,  y^l,  y^y^  and  y^z\\  and  then  the  terms  which  con- 
tain the  second  power  of  ylt  viz.,  y\yl  and  y\z^  and,  finally,  least 
important  of  all  for  this  particular  case,  the  term  which  contains 
y\.  In  the  developments  of  the  aberrations  of  the  3rd  order,  where 
ylt  zl  =  o,  yl  and  zt  denote  the  approximate  values  of  the  parame- 
ters of  the  object-ray,  the  ten  terms  above-mentioned  are  all  that  can 
occur. 


376  Geometrical  Optics,  Chapter  XII.  [  §  260. 

The  expressions  for  the  aberrations  dy'm,  dzni,  which  are  developed 
by  SEIDEL  (see  Art.  102),  enable  us  to  compute  the  resultant  defects 
of  the  3rd  order  of  the  image  of  an  object-point,  and  by  specializing 
these  general  formulae  (as  SEIDEL  himself  does),  we  can  ascertain  the 
nature  of  the  various  component  defects  which  go  to  make  up  this 
resultant.  However,  in  order  to  obtain  a  clear  comprehension  of  these 
errors,  it  is  best  to  follow  the  plan  adopted  by  KOENIG  and  VON  RoHR1  in 
their  admirable  and  exhaustive  treatise  on  the  Theory  of  Spherical  Aber- 
rations, and  thus,  first,to  develop  separately  the  formulas  for  each  one 
of  these  special  aberrations,  and  afterwards  to  give,  at  the  end  of  the 
chapter,  SEIDEL'S  general  theory  (Arts.  102,  foil.).  Accordingly,  this 
method  will  be  pursued  here  also. 

II.     THE  SPHERICAL  ABERRATION  IN  THE  CASE  WHEN  THE  OBJECT-POINT  LIES  ON  THE 

OPTICAL  Axis. 

ART.  81.     CHARACTER   OF   A   BUNDLE    OF   REFRACTED   RAYS   EMANATING 
ORIGINALLY  FROM    A   POINT    ON   THE    OPTICAL   AXIS. 

260.  Longitudinal  Aberration,  or  Aberration  along  the  Optical 
Axis.  The  simplest  case  of  all  is  the  case  when  the  object-point  lies 
on  the  optical  axis  of  the  centered  system  of  spherical  surfaces,  so 
that  the  point  designated  by  Pl  coincides  with  Mlt  that  is,  yl  =  o. 
When  the  bundle  of  image-rays  is  symmetrical  about  an  axis,  as  is  the 
case  when  the  rays  emanate  originally  from  a  point  on  the  optical  axis, 
one  of  the  caustic  surfaces  (§  46)  is  a  surface  of  revolution  around  the 
axis  and  is  touched  by  each  ray  of  the  bundle;  whereas  the  other  caustic 
surface,  in  this  particular  instance,  collapses  into  the  segment  of  the 
axis  comprised  between  the  point  M'  (Fig.  133)  where  the  paraxial 
rays  cross  the  axis  and  the  point  L'  where  the  outermost  rays  of  the 
bundle  meet  the  axis.2  All  the  rays  of  the  bundle  will  intersect  the 
axis  at  points  which  are  comprised  between  the  two  extreme  points 
M'  and  L'.  This  axial  line-segment  Mf L'  is  called  the  Longitudinal 
Aberration  of  the  outermost  ray.  Let  A  designate  the  vertex  of  the 
spherical  surface,  and  let  us  put  AM'  =  u',  AL'  =  v'.  If  the  Spheri- 

1  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen;  being 
Chapter  V  (pages  208-338)  of    Die  Theorie   der  optischen  Instrumente,  Bd.  I,  edited   by 
M.  VON  ROHR  (Berlin,  1904).     This  treatise  of  Messrs.  KOENIG  and  VON  ROHR  has  been 
of  inestimable  service  to  the  author  in  the  preparation  of  the  present  chapter  of  this 
work. 

2  All  the  letters  in  the  figure  should,  as  a  matter  of  fact,  be  written  with  the  subscript 
m,  to  indicate  that  the  letters  relate  to  the  rays  after  refraction  at  the  last,  or  mth,  surface 
of  the  system.     But  in  all  such  cases  as  the  one  here  considered  the  surface-numerals 
written  as  subscripts  may  be  conveniently  omitted  where  only  one  of  the  surfaces  of  the 
system  is  being  treated,  since  there  is  no  risk  of  confusion. 


§  260.] 


Theory  of  Spherical  Aberrations. 


377 


cal  Aberration  along  the  axis,  or  the  Longitudinal  Aberration  M' L' 
is  denoted  by  du',  we  shall  have : 

5ur  =  v'  -  u'. 

Now  if  0'  =  /.AL'B  denotes  the  slope  of  the  ray  which,  in  the  plane 
of  the  diagram,  crosses  the  optical  axis  at  the  point  designated  by  L', 
it  is  evident  that  5uf  is  a  function  of  this  angle  0';  and,  moreover,  it 
is  also  evident  that  if  the  function  du'  is  developed  in  a  series  of  as- 


FIG.  133. 

CHARACTER  OF  BUNDLE  OF  RAYS  SYMMETRICALLY  SITUATED  WITH  RESPECT  TO  THE  OPTICAL 

Axis. 


cending  powers  of  6',  only  the  even  powers  will  occur,  because  for  a 
ray  lying  in  the  same  meridian  plane  and  symmetrically  situated  on 
the  other  side  of  the  optical  axis,  so  that  its  slope-angle  is  equal  to 
—  0',  we  shall  obtain  the  same  value  of  the  function  du'.  If,  there- 
fore, 0'  may  be  regarded  as  a  magnitude  of  the  ist  order  of  smallness, 
we  can  write  : 


du' 


(273) 


since  all  the  succeeding  terms  of  the  series,  involving  magnitudes  of 
the  orders  of  smallness  higher  than  the  3rd,  are,  by  the  limitations 
of  this  investigation,  to  be  neglected. 

The  co-efficient  a'  is  entirely  characteristic  both  of  the  magnitude 
and  of  the  nature,  or  sign,  of  the  aberration  du',  since,  for  a  given 
value  of  0',  we  can  determine  du',  so  soon  as  we  have  ascertained 
also  the  value  of  a'.  Thus,  if  a!  =  o,  we  have  vr  =  ur,  in  which  case 
we  say  that  the  system  is  "spherically  corrected"  for  this  ray.  Ac- 


378  Geometrical  Optics,  Chapter  XII.  [  §  261. 

cording  as  the  sign  of  the  co-efficient  a'  is  positive  or  negative,  the 
optical  system  is  said  to  be  "spherically  over-corrected"  or  "spherically 
under-corrected"  for  the  particular  ray  in  question. 

261.  Least  Circle  of  Aberration.  If  the  rays  of  the  bundle  are 
received  on  a  plane  screen  placed  at  right  angles  to  the  optical  axis, 
and  if  this  screen  is  gradually  translated  along  the  axis  from  the  posi- 
tion G'H'  in  the  figure  towards  the  point  Mf ',  we  shall  see  on  the 
screen  at  first  a  circular  patch  of  light  surrounded  on  its  outer  edge 
by  a  brighter  ring,  which  will  gradually  contract  as  the  screen  ap- 
proaches the  point  L',  where  now  the  effect  of  the  other  caustic  will 
begin  to  be  manifest,  and.  as  the  screen  is  advanced  still  farther 
from  L'  towards  M' ,  we  shall  see  at  the  centre  of  the  circular  patch 
of  light  an  increasingly  bright  spot.  A  plane  perpendicular  to  the 
optical  axis  at  the  point  Nf  in  the  figure  will  meet  the  outside  rays 
of  the  bundle  at  the  points  where  these  rays  cross  the  caustic  sur- 
face of  revolution,  and  in  this  plane  we  shall  evidently  have,  there- 
fore, the  narrowest  contraction  of  the  bundle  of  rays.  By  some  writers 
on  Optics  the  circle  of  light  which  appears  on  the  screen  when  it  is 
placed  at  N'  is  called  the  Least  Circle  of  Aberration. 

Let  h  =  DB  denote  the  incidence-height  at  the  last  spherical  surf- 
ace of  the  extreme  ray  of  the  bundle,  whose  slope-angle  is 

0'  =  Z.AL'B\ 

and  let  L"  (not  shown  in  the  figure)  mark  the  position  of  the  point 
between  L'  and  M' ',  where  some  other  ray  of  the  pencil  of  image-rays 
lying  in  the  plane  of  the  figure  crosses  the  optical  axis.  The  slope- 
angle  of  this  ray  may  be  denoted  by  6".  Finally,  let  i  denote  the 
ordinate  of  the  point  of  intersection  of  this  general  ray  with  the  outer- 
most ray.  The  least  value  i0  of  i  will  be  the  radius  of  the  Least 
Circle  of  Aberration.  Evidently, 

L"L'  =  i(cot0"  -cot  0'); 
moreover,  if  a'  denotes  the  so-called  aberration-co-efficient  (§  260) : 

L"Lf  =  a'(0'2-0"2): 
so  that  we  obtain  the  following  relation : 

*(cot0"  -  cot  00  =  a'(0'2  -  0"2). 
Differentiating  this  equation  with  respect  to  0",  and  putting  di/dO"=o, 


§  262.]  Theory  of  Spherical  Aberrations.  379 

we  obtain: 


''  -  cot  0')  =  0'   -  0"2; 

and  expanding  the  trigonometric  functions  in  series,1  and  neglecting 
terms  involving  powers  and  products  of  0',  0"  higher  than  the  3rd, 
we  find: 

0"(20"  -  00  =  0'2, 

which  is  satisfied  by  the  values  0"  =  0'  and  0"  =  -  0'/2.  The  first 
of  these  values  corresponds  to  the  maximum  value  of  i  represented  in 
the  diagram  by  the  ordinate  G '  H'\  whereas  the  second  value 

0"  =  _  0'/2 

gives  the  slope  of  the  ray  for  which  i  =  i0  is  a  minimum.  The  inci- 
dence-height of  this  ray  is,  therefore,  approximately  half  that  of  the 
outside  ray,  but  opposite  in  sign.  If  this  value  of  0"  is  substituted  in 
the  above  equation  connecting  i  and  0",  we  shall  find  (neglecting,  as 
before,  powers  of  0'  above  the  3rd)  for  the  radius  of  the  Least  Circle 
of  Aberration: 

i,  =  -  a'0'3/4. 

The  position  on  the  axis  of  the  point  AT7  can  be  determined  from 
the  fact  that  N'  Lf  must  be  equal  to  —  vcot  0'l  hence,  to  the  same 
degree  of  approximation,  we  find : 

N'L'  =  a'0'2/4  =  M'L'l\. 

Accordingly,  the  distance  of  the  Least  Circle  of  Aberration  from  the 
GAUSsian  Image-Point  M'  is  equal,  approximately,  to  three-fourths 
of  the  Longitudinal  Aberration  of  the  extreme  outside  ray. 

262.  The  so-called  Lateral  Aberration.  Exactly  what  point  on 
the  axis  is  to  be  regarded  as  the  image  of  the  axial  object-point  M  in 
such  a  case  as  that  which  we  are  here  discussing  is  a  question  that 
cannot  be  decided  by  merely  theoretical  considerations;  especially, 
too,  as  there  is  some  diversity  of  opinion  on  the  subject.  In  order  to 
be  answered,  the  matter,  as  CZAPSKI  observes,  needs  to  be  considered 
rather  from  the  point  of  view  of  Physical  Optics  than  from  that  of 
Geometrical  Optics.  Most  optical  writers  are  agreed,  however,  that 
the  place  probably  selected  by  the  eye  as  most  nearly  reproducing  the 
axial  object-point  is  the  place  of  the  least  circle  of  aberration.  This 

1  The  development  of  the  cotangent  in  series  is  as  follows: 

cot  *  =  i  /*  —  */3  —  *3/45  —  2X5/94S • 


380  Geometrical  Optics,  Chapter  XII.  [  §  263. 

circle  is  pierced  by  all  the  rays  of  the  bundle,  and  the  radius  of  it 
might  be  considered  as  the  measure  of  the  spherical  aberration.  In- 
stead of  this  magnitude,  ABBE  employs  the  radius  of  the  circle  in  the 
GAUSsian  Image-Plane  <r'  (§  254),  inside  of  which  all  the  rays  of  the 
bundle  cross  this  plane.  This  radius 

M'V  =  8yf 

is  called  the  Lateral  Aberration  of  the  extreme  ray,  and  its  magnitude 
is  equal  to  M'L' /N'L'  times  the  radius  of  the  least  circle  of  aberration: 
that  is, 

$/  =  -o'.fa'  =  -  a'-B\  (274) 

Thus,  we  see  also  that,  whereas  the  Longitudinal  Aberration  5u'  is  of  the 
2nd  order  of  smallness,  the  Lateral  Aberration  by'  is  of  the  3rd  order. 

ART.  82.     DEVELOPMENT  OF  THE   FORMULA  FOR  THE   SPHERICAL  ABER- 
RATION OF  A  DIRECT  BUNDLE  OF  RAYS. 

263.  Since  the  bundle  of  rays  emanating  originally  from  a  point  on 
the  optical  axis  of  the  centered  sytem  of  spherical  surfaces  is  sym- 
metrical with  respect  to  this  axis,  it  will  be  sufficient  to  investigate 
the  rays  in  any  meridian  plane.  Consider,  therefore,  any  ray  of  the 
bundle,  and  let  the  meridian  plane  containing  this  ray  be  the  xy- 
plane  of  the  system  of  co-ordinates.  Hence,  for  this  ray  not  only  do 
we  have  3^  =  zt  =  o  (as  is  the  case  for  all  the  rays  of  the  bundle), 
but  also  ZL  =  o;  so  that  the  only  term  in  the  series-development  of 
the  Lateral  Aberration  dy'm  will  be  the  .yj-term  (see  §  259). 

Discarding  for  the  present  the  subscript-notation,  let  us  designate 
by  L,  L'  the  points  where  the  path  of  this  ray  crosses  the  optical  axis 
before  and  after  refraction  at  the  &th  spherical  surface.  Employing 
here  the  same  letters  and  symbols  as  in  §  209,  viz. : 

r  =  AC,    v  =  AL,    v'  =  AL't     Z.BCA=<p,     ^ALB=  6, 

^AL'B  =  0', 

denoting  also  the  angles  of  incidence  and  refraction  by  a,  af,  respect- 
ively, and  the  indices  of  refraction  by  n,  n' ,  we  may  write  the  funda- 
mental formulae  for  the  refraction  of  the  ray  at  the  spherical  surface 
in  question,  as  follows  (see  §  210) : 

r  •  sin  a.  =  —(v  —  r)-  sin  6,  " 


r  -  sin  a'  =  —  (v'  —  r)  -  sin  0', 
/  •  sin  a.'  =  n  •  sin  a, 
a  -*  e  =  a'  -  8'  =  <p. 


(275) 


§  263.]  Theory  of  Spherical  Aberrations.  381 

If,  also,  M,  M'  designate  the  points  where  the  path  of  a  paraxial  ray 
crosses  the  optical  axis,  before  and  after  refraction,  respectively,  at 
the  kth  spherical  surface,  and  if 

AM  =  u,     AM'  =  «', 
then  (§126) 


Moreover, 

ML  =  8u  =  v  -  u,     M'L'  =  du'  =  v'  -  u' 

will  denote  the  magnitudes  of  the  Longitudinal  Aberration  of  the  ray 
before  and  after  refraction  at  the  spherical  surface. 

If  we  neglect  all  magnitudes  higher  than  those  of  the  second  order, 
the  approximate  values  of  the  slope-angles  6,  6',  expressed  in  terms 
of  the  central  angle  <p,  are  6  =  —  rp/u,  6'  =  —  r<p/u'.  But  if,  as  we 
propose  to  do  here,  we  take  account  also  of  the  terms  of  the  3rd  order, 
the  expressions  for  0,  0'  must  evidently  have  the  following  forms: 


0=  - 


(276) 


wherein  the  co-efficients  A,  A'  are  undetermined.     Moreover,  since 

a  =  9  +  <p,     a'  =  6'  +  <p, 

we  may  expand  a,  a'  likewise  in  a  series  of  odd  powers  of  <p,  as  follows: 

Jr 

<P  +  AV, 

(277) 


a  =  - 
n 


where  the  co-efficients  A,  A'  have  the  same  meanings  as  in  formulae 

(276). 

If  x  denotes  a  small  magnitude  of  the  1st  order,  and  if  we  take 
account  of  terms  as  far  as  #3,  then 

sin  x  =  x  —  x3/6; 
and,  hence,  employing  formulae  (276)  and  (277),  we  have  here  the  fol- 


382  Geometrical  Optics,  Chapter  XII.  [  §  263. 

lowing  series-developments  for  the  sines  of   the   angles  a,  a',  6,  6': 


Jr 

sin  a  —  — 

Jr 


i  JV 

i  7V3 


(278) 


Substituting  in  the  first  three  of  equations  (275)  these  values  of  the 
sines  of  the  angles  a,  0,  etc.,  putting 

v  =  u  +  du,     v'  =  u'  +  8u', 

and  neglecting  all   magnitudes  of  orders  higher  than  the  3rd,  we 
obtain,  after  some  reductions: 


Su          v*(Jr*(J      i\  u} 

—  =  —  ~r  {  —  i  ---  I  —  6A  -  \  , 
u  6    [  n    \n      u  )  r  J 


u'  ~        6\n'  \n' 
6(n'A'  -  nA)  = 


(  -^  -  -A 
\n        n  ) 


If,  now,  we  multiply  the  first  of  these  equations  by  n/u  and  the  second 
by  n'  /u',  and  then  subtract  the  first  from  the  second,  we  obtain: 


du 

~2 
IT 


/  i          i\     /i       i\       6  ,  1N1 

I  -r~;  r  ---  )-(^2-"-2)~Ti(w^  -w^4)f  ; 
\wV      ww/     \«       IT/     /r 


and,  accordingly,  by  means  of  the  third  of  the  equations  above,  we 
can  eliminate  at  the  same  time  both  of  the  unknown  co-efficients  A,  Af. 
Thus,  employing  ABBE'S  convenient  Difference-Notation,  whereby  the 
difference  q'  —  q  between  the  values  q,  qf  of  a  magnitude  before  and 
after  refraction  is  denoted  by  Ag,  we  derive  the  following  equation: 


n-8u 

—  2" 

u 


nu 


72 

—  A—  —  J  - 


u 


i  \ 

~2  J, 

n  J 


UNIVERSITY 

OF 
FO 

§  264.]  Theory  of  Spherical  Aberrations.  383 

which,  since  (§126) 


n       r     n         nu 
and 


u  \    nu      r     n 

may  be  still  further  simplified  as  follows: 

n  -  du  1  2  2  2       i 

A  — 2~~  =  ~  kr  V  J  '**  —  •  (279) 

IT  WW 

Thus,  provided  we  know  the  Longitudinal  Aberration  8u  of  the  given 
ray  before  refraction  at  the  spherical  surface  in  question,  we  may,  by 
means  of  formula  (279),  compute  the  magnitude  du'  of  the  Longitudi- 
nal Aberration  after  refraction. 

264.  From  the  incidence-point  B  draw  BD  perpendicular  to  the 
optical  axis  at  B,  and  put  DB  =  h,  so  that  h  denotes  the  ordinate  of 
the  incidence-point  B,  that  is,  the  incidence-height  of  the  ray  of  slope 
6.  Then,  since 

h  =  r-sin  <p, 

to  the  degree  of  approximation  required  in  this  investigation,  we  may 
write : 

3 

h  =  r<p  —  r  -r ;  (280) 

and,  consequently,  in  formula  (279)  we  can  put  r2p2  =  h2.  If  we  do 
this,  and  if  now  at  the  same  time  we  attach  to  the  symbols  the  surface- 
number  in  the  form  of  a  subscript,  noting  also  that  the  point  Lk  where 
the  ray  crosses  the  axis  before  refraction  at  the  &th  surface  is  identical 
with  the  point  L[._\  where  the  ray  crosses  the  axis  after  refraction  at 
the  (k  —  i)th  surface,  so  that 


we  may  write  the  formula  for  the  kth  surface  as  follows: 


Uk 

or  in  ABBE'S  abbreviated  notation  : 


384  Geometrical  Optics,  Chapter  XII.  [  §  265. 

265.  By  means  of  this  recurrent  formula  (281),  we  can  obtain 
finally  the  value  of  the  Longitudinal  Aberration  5um  of  the  ray  after 
refraction  at  the  last  surface  of  the  centered  system  of  m  spherical 
surfaces.  How  this  is  done,  we  proceed  now  to  show. 

By  combining  formulae  (276)  and  (280),  we  find: 


(AU  - 


h  =  -  ud  +  I  Au  -  7  I  <f  =  -  iiV  +    A'u'  -  7  1  ^  ,      (282) 
\  o/  \  o/ 

and,  hence,  again  introducing  the  subscripts,  and  remarking  that  the 
angles  denoted  by  6k  and  ^._L  are  identical,  we  may  write,  taking 
account  of  the  terms  of  the  3rd  order: 


If,  therefore,  we  multiply  both  sides  of  equation  (281)  by  h2k,  and 
at  the  same  time  use  the  relation  (283),  we  shall  derive  the  following 
formula  : 


U 


/  i  \ 
I  rr  I  - 

\  nu  /  k 


If,  now,  in  this  formula  we  give  k  in  succession  the  values  1,2,  •  •  •  m, 
and  add  together  the  equations  thus  obtained,  and  note  also  that, 
since  the  bundle  of  object-rays  is  supposed  to  be  homocentric,  we  must 
put  5Ui  =  o,  we  obtain  finally: 


'2     74  h=m  /  I,    \4  /     T     \ 

-A  art?  i  Ji-*(  -)  • 

zn»kl£l\kj  \nujk 


In  this  formula  we  need  to  know,  in  addition  to  the  constants 
which  determine  the  optical  system  (refractive  indices,  radii,  thick- 
nesses, etc.),  only  the  position  on  the  axis  of  the  object-point  Ml  and 
the  incidence-height  /^  of  the  object-ray;  for  then  we  can  compute 
the  values  of  all  the  other  magnitudes  that  occur  on  the  right-hand 
side  of  the  equation.  Practically,  the  formula  is  very  convenient, 
because  it  exhibits  the  effect  on  the  Longitudinal  Aberration  bu'm  which 
is  produced  at  each  refraction.  For  a  given  axial  object-point,  it  will 
always  be  theoretically  possible,  by  employing  a  sufficient  number  of 
surfaces,  to  contrive  so  that  the  aberration  8um  =  o;  the  condition 
whereof  is: 


(285) 

k=\ 


§  266.]  Theory  of  Spherical  Aberrations.  385 

It  must  be  remembered,  however,  that  the  accuracy  of  this  formula 
for  the  abolition  of  the  spherical  aberration  along  the  axis  depends  on 
the  magnitude  of  the  aperture  of  the  bundle  of  rays;  for  it  has  been 
assumed  throughout  that  we  can  safely  afford  to  neglect  the  powers 
of  the  slope-angle  6  higher  than  the  3rd.  Thus,  for  example,  in  the 
case  of  the  objective  of  a  telescope,  the  aperture  of  which,  although 
by  no  means  negligible,  is  relatively  small,  the  formula  will  usually 
give  a  very  high  approximation.  On  the  other  hand,  in  the  calcula- 
tion of  a  photographic  objective  the  formula  would  generally  not  be 
very  accurate.  In  the  objective  of  a  microscope  the  magnitude  of 
the  angle  6  is  often  equal  to  nearly  90°,  and  the  approximate  formulae 
here  derived  are  not  applicable  to  wide-angle  systems  at  all. 

266.  Abbe's  Measure  of  the  "Indistinctness"  of  the  Image.  By 
means  of  formulae  (.274)  and  (282),  we  find  for  the  Lateral  Aberration: 


and  hence: 


If  e^k  denotes  the  length  of  the  object-line  perpendicular  to  the 
optical  axis  at  Mlt  whose  GAUSsian  image  at  M'k  is  equal  to  the  value 
of  the  Lateral  Aberration  5y'k  after  refraction  at  the  kth  surface,  it  is 
evident  that  details  in  an  object  at  Ml  which  are  separated  by  an  inter- 
val greater  than  e\^  will,  on  account  of  the  spherical  aberration,  not 
appear  separated  in  the  image  formed  after  refraction  at  the  kth 
surface.  Thus,  according  to  ABBE,  the  magnitude  denoted  by  e\.^ 
measured  at  the  object,  affords  a  convenient  measure  of  the  lack  of 
detail,  or  "indistinctness",  of  the  image. 

The  approximate  value  of  the  slope-angle  B'k  is: 


nk+\ 


and,  hence  by  the  Law  of  ROBERT  SMITH  (§  194) 


Wk    "Jk 

«1  Uk 

that  is, 


26 


n'A.'8yk 

t 

(287) 


386  Geometrical  Optics,  Chapter  XII.  [  §  267. 

Thus,  from  formula  (286)  we  obtain: 

*---i«*S  (£)'**(£).'          (288) 

which  shows  that  the  "indistinctness"  is  proportional  to  the  cube  of 
the  aperture  hv  of  the  bundle  of  object-rays. 

In  case  the  object-point  Ml  is  very  far  away,  it  will  be  convenient 
to  determine  the  angle  e1>m  subtended  at  the  vertex  Al  of  the  first 
surface  by  the  linear  magnitude  £i,m;  thus,  since 

€l,m  =  Cl.m/Ui, 

we  have: 


and,  hence,  the  angular  value  of  the  lack  of  detail  in  the  image,  on 
account  of  spherical  aberration,  is  proportional  to  the  cube  of  the 
linear  aperture  /^  of  the  bundle  of  object-rays.  For  example,  in  the 
case  of  the  objective  of  a  telescope,  it  is  proportional  to  the  cube  of  the 
diameter  of  the  objective. 

ART.  83.     SPHERICAL  ABERRATION   OF  DIRECT  BUNDLE   OF  RAYS  IN 

SPECIAL  CASES. 

267.     Case  of  a  Single  Spherical  Refracting  Surface. 

If  the  optical  system  consists  of  a  single  spherical  surface  (m  —  i), 
we  have  for  the  Longitudinal  Aberration  of  the  bundle  of  image-rays 
corresponding  to  a  bundle  of  object-rays  proceeding  from  the  axial 
point  M  (u  =  AM)  : 

T27.2      '2 


JVuf  i        i  \ 

buf  = r-  1  -r-f I ; 

2n     \nu      nu  J 


and  if  we  substitute  for  J  its  value,  viz.  : 

n'(u'  -  r) 


_ 


ru 
we  obtain: 

h\u'  -  rf(nu  -  »  V) 

Su  =  -        ~  (289) 


In  each  of  the  following  three  cases  the  Longitudinal  Aberration  will 
be  equal  to  zero  : 

(i)  When  u  =  uf  =  o,  in  which  case  object-point  M  and  image- 
point  M'  coincide  at  the  vertex  A  of  the  sphere; 


I 

§  268.]  Theory  of  Spherical  Aberrations.  387 

(2)  When  u  =  u'  =  r,  in  which  case  object-point  M  and  image- 
point  Mr  coincide  at  the  centre  C  of  the  sphere;  and 

(3)  When  nu  =  n'u',  in  which  case: 

u  =  (n  +  n')rjn,     u'  =  (n  +  n')r/n', 

and  the  points  M,  M'  coincide  with  the  aplanatic  points  Z,  Z',  re- 
spectively (§  207,  §  211,  Note  3).  This  latter  case  is  the  only  one  that 
may  be  said  to  have  any  practical  importance. 

For  all  other  positions  of  the  object-point  M  the  point  Lf  will  not 
coincide  with  M' '.     The  sign  of  5u'  will  depend  on  the  sign  of  the  factor 


i          i     _n  —  n  f\      n  +  n    i\ 
'u       nu         n'2     \  r          n       u ) ' 


nu 

and  for  any  given  spherical  surface  may  be  positive  or  negative  de- 
pending on  the  sign  of  u.  If  the  object-point  M  is  at  an  infinite 
distance,  the  sign  of  5u'  will  depend  on  that  of  (n'  —  n)/r.  If  this 
expression  is  positive,  the  refracting  surface  will  be  a  convergent  sur- 
face, the  sign  of  5u'  will  be  negative,  and  the  spherical  surface  will  be 
"spherically  under-corrected"  (§  260). 

268.     Case  of  an  Infinitely  Thin  Lens. 

When  the  optical  system  consists  of  two  spherical  surfaces,  we  must 
put  m  =  2  in  formula  (284).  Assuming  that  the  Lens  is  surrounded 
by  the  same  medium  on  both  sides,  we  may  conveniently  write: 

n  =  n'Jn,  =  n[/n'2, 

so  that  in  the  following  discussion  n  will  be  used  to  denote  the  rela- 
tive index  of  refraction  for  the  two  media  concerned.  Moreover,  in 
the  case  of  an  Infinitely  Thin  Lens,  we  have: 

u[  =  u2, 

and  we  may  therefore  afford  to  dispense  with  the  subscripts  in  the 
symbols  u^  and  u'2,  and  write  these  u  and  u,  respectively.  Likewise, 
we  shall  write:  h  =  hY  =  h2.  Under  these  circumstances,  we  obtain 
by  the  general  formula  (284)  the  following  expression  for  the  Longitu- 
dinal Aberration  of  an  Infinitely  Thin  Lens: 

(290) 

For  the  case  of  an  Infinitely  Thin  Lens  we  shall  employ  a  special 
notation,  as  follows:  Thus,  let  x  =  i/u,  x'  —  i/u'  denote  the  recip- 


388  Geometrical  Optics,  Chapter  XII.  [  §  268. 

rocals  of  the  intercepts  on  the  axis  of  the  paraxial  object-rays  and 
image-rays,  respectively,  and  let  c  =  i/rlt  c'  =  i/r2  denote  the  curv- 
atures of  the  bounding  surfaces  of  the  Lens.  Finally,  let  p  =  iff 
denote  here  the  reciprocal  of  the  primary  focal  length  of  the  Lens. 
With  this  system  of  symbols  the  formulae  of  Chapter  VI,  Art.  41, 
for  the  Refraction  of  Paraxial  Rays  through  an  Infinitely  Thin  Lens 
will  have  the  following  forms: 


<p  =  (n  —  \](c  -  c), 

x  =  x  +  <p, 

j       x  +  (n  —  i)c 


n 


(291) 


Employing  these  relations,  we  can  eliminate  from  formulas  (290)  the 
magnitudes  denoted  by  ttt,  x'  and  c',  and  thus  we  shall  obtain: 


i  (c  -*){(»  +  2)  (c  -x)-  znx  }], 


or 

a*'  =  -^P 


+  ^_CT  +  <?.      (292) 

If  the  object-rays  are  parallel  to  the  axis  (x  =  o,  x'  =  <p),  the  image- 
point  Jlf  coincides  with  the  secondary  focal  point  £',  and  for  this 
special  case  we  obtain: 


EL'  =  - 


In  the  case  of  a  Thin  Lens  of  semi-diameter  h  and  focal  length/,  whose 
thickness  is  greatest  along  the  optical  axis,  one  can  easily  see  from 
the  geometrical  properties  of  the  circle  that  the  thickness  of  the  Lens 
is  very  nearly  equal  to 


and  thus  for  a  Lens  of  this  character  the  expression  within  the  large 


§  269.]  Theory  of  Spherical  Aberrations.  389 

brackets  of  formula  (293)  is  the  factor  by  which  the  thickness  of  the 
Lens  has  to  be  multiplied  in  order  to  obtain  the  spherical  aberration 
along  the  axis  for  an  infinitely  distant  axial  object-  point.  If  the  Lens 
is  a  convergent  glass  Lens  in  air  (n  =  3/2),  its  thickness  is  very  nearly 
equal  to  h2/f. 

By  way  of  illustration,  let  us  compute  by  formula  (293)  the  Longi- 
tudinal Aberration  for  Lenses  of  special  forms;  thus,  we  shall  find: 

(i)   In  case  the  first  surface  of  the  Lens  is  plane  (c  =  o)  : 


(2)  In  case  the  second  surface  of  the  Lens  is  plane  (c  —  v/(n  —  i)): 

Vym  -*=*£+*.*.    for    ,_3/a,     B'L'.-Zf 

n(n-  i)2      2/f  6  / 

(3)  In  case  of  an  Equi-Biconvex  Lens  (c  =  —  c'  =  vJ2(n  —  i)): 


Assuming,  therefore,  that  the  focal  length  /  of  each  of  these  Lenses  has 
the  same  numerical  value,  we  see  that  the  Longitudinal  Aberration  is 
greatest  in  the  Lens  with  its  plane  side  turned  towards,  and  least  in 
the  Lens  with  its  plane  side  turned  away  from,  the  object-rays. 

269.  The  next  question  to  be  investigated  is,  What  are  the  conditions 
that  the  Longitudinal  Aberration  of  a  Thin  Lens  shall  vanish  ? 

If,  for  brevity,  the  expression  within  the  large  brackets  in  formula 
(292)  is  put  equal  to  Z,  we  may  write  the  formula  for  the  Longitudinal 
Aberration  of  a  Thin  Lens  as  follows  : 

•     te,=  _£4?z.  ,        (294) 


If  (p  =  o  (that  is,/  =  oo  ),  we  shall  have  du'  =  o.  In  this  case  u  =  u'  , 
rl  =  r2,  so  that  the  two  surfaces  of  the  Infinitely  Thin  Lens  are  parallel. 
This  case  has  evidently  no  practical  interest.  It  remains,  therefore, 
to  investigate  the  cases  when  the  function  Z  vanishes. 

We  shall  assume  that  we  have  given  a  Lens  of  a  definite  focal 
length,  and  that  the  position  on  the  axis  of  the  object-point  M  is  also 
given;  and,  under  these  circumstances,  we  are  required  to  determine 
the  form  of  the  Lens  in  order  that  the  Longitudinal  Aberration  shall 
be  zero;  that  is,  we  must  ascertain  the  curvatures  c,  c'  of  the  two  surf- 
aces of  the  Lens.  Since  c'  =  c  -f  (p,  and  since  the  value  of  <p  is  sup- 
posed to  be  prescribed,  the  problem,  in  reality,  consists  merely  in 


390  Geometrical  Optics,  Chapter  XII.  [  §  269. 

finding  the  curvature  c  of  the  first  surface.  This  process  of  varying 
the  curvatures  of  the  surfaces  without  altering  the  focal  length  is 
called  "bending"  the  Lens. 

Accordingly,  regarding  c  as  the  independent  variable,  and  treating 
both  x  and  <p  as  constants,  we  shall  write  the  function  Z  in  the  follow- 
ing form: 


J  ,2n+i 

c  —  I  -x  +  -     —  <p 

n  \       n  n  -  I  * 

3^ 


For  Z  =  o,  we  obtain  two  values  of  c,  as  follows : 

.2 


_  4(w2-  i)x  +  w(2w  +  i)<p±n     ±(n  -  i)2x(x  +  <p)  —  (4»  -  i)^2 

2(»  -i)(»+  2) 

and  if  these  values  of  c  are  to  be  real,  the  expression  under  the  radical 
must  be  positive;  that  is,  for  real  values  of  c,  we  must  have: 


or,  snce  x  +  <f>  = 


Accordingly,  we  see  that  a  necessary  condition  that  the  aberration 
shall  vanish  is  that  x  and  x'  shall  have  the  same  sign;  which  means 
that  the  object-point  and  image-point  must  lie  both  on  the  same  side 
of  the  Lens.  In  the  practical  and  more  important  case  when  the  image 
is  a  real  image,  it  is  impossible  to  abolish  the  Longitudinal  Aberration 
of  an  Infinitely  Thin  Lens. 

The  condition  above  may  also  be  put  in  the  following  form: 


\n  —  i  x  \n  —  i 

Thus,  with  a  glass  Lens  in  air  (n  —  3/2),  it  is  possible  to  abolish  the 
Longitudinal  Aberration  only  in  case  the  ratio  x'  jx,  or  u' /u,  is  com- 
prised between  the  values  (n  —  l/2i)/io  and  (n  +  l/2i)/io,  that 
is,  between  the  values  0.642  and  1.558. 

In  exactly  the  same  way,  by  considering  Z  as  a  function  of  x,  and 
treating  c  and  <p  as  constants,  we  shall  find  that  in  order  for  Z  to 


§  270.]  Theory  of  Spherical  Aberrations.  391 

vanish  for  real  values  of  x,  the  Infinitely  Thin  Lens  must  have  a  form 
such  that 

c2  _       i      c  _  (n  +  0(3*  ~  0  > 

<pz     n  —  i  (p  \(n  —  i)2 

that  is,  the  ratio  c/(p  must  be  comprised  between  the  values 


and 


\  ,          v        . 

2(n  —  i)  2(n  —  i) 

For  example,  for  n  =  3  /2,  the  value  of  c/<p  must  lie  between  (2  —  1/39)  /2 
and  (2  +  1/39)  /2  that  is,  between  —  2.1225  and  +  4.1225. 

Practically  speaking,  these  results  are  without  value. 

270.  Since,  therefore,  it  is  practically  not  feasible  to  abolish  en- 
tirely the  Longitudinal  Aberration  in  the  case  of  an  Infinitely  Thin 
Lens,  let  us  seek  now  to  find  the  condition  that  the  Aberration  shall 
be  a  minimum. 

Equation  (295),  in  which  c  and  Z  are  to  be  considered  as  the  vari- 
ables, evidently  represents  a  Parabola  with  its  axis  parallel  to  the 
Z-axis  of  co-ordinates  and  with  its  vertex  at  the  point: 


n 


f 

4(n  —  i}(n  +  2)*       n  +  2 

2(n  -f  i)  n(2n-\-  i) 

n  +  2        f  2(«  -i)(»  +  2) 


(296) 


and  it  is  obvious  that  for  a  Lens  of  given  "power"  (<p)  and  fora  given 
position  (x)  of  the  object-point  M  on  the  axis,  the  minimum  value  of 
the  function  Z  will  be  Z  =  Z0. 

So  long  as  xx'  =  x(x  +  <p)  is  not  positive,  the  value  of  Z0  cannot 
be  equal  to  zero;  if  xx'  <;  o,  then  Z0  >  o.  That  is,  for  a  real  image- 
point  M'  on  the  other  side  of  the  Lens  from  the  object-point  M,  the 
minimum  value  of  Z  is  positive.  In  case  xx'  >  o,  Z0  will,  in  general, 
be  negative,  and  in  special  cases  it  may  be  equal  to  zero,  in  agreement 
with  the  results  found  in  the  preceding  discussion.  We  need  consider 
only  the  case  when  Z0  >  o.  The  minimum  value  of  the  Longitudinal 
'Aberration  is: 


2 

For  an  infinitely  distant  object-point  (x  =  o,  x'  =  ^>),  the  curvatures 


392  Geometrical  Optics,  Chapter  XII.  [  §  271. 

of  the  Lens-surfaces  for  minimum  aberration  are: 

n(2n  +  i)  ,          2w2  —  n  —  4 

and  the  minimum  aberration  is: 


For  n  =  3/2,  we  find:  c0  =  12^/7,  (E'L')0  =  —  15^/14;  and  for 
w  =  2  (diamond),  c0  =  5^/4,  (E'L')0  =  —  7h2<p/i6.  The  minimum 
value  of  the  Longitudinal  Aberration  of  a  Diamond  Lens  is  very  much 
less  than  that  of  a  Glass  Lens  of  equal  focal  length.  And,  generally, 
for  values  of  n  greater  than  unity,  it  is  easy  to  show  that  the  minimum 
value  of  the  aberration  decreases  with  increase  of  n. 
When  x  =  o,  we  have  : 

,         n(2n+  i) 


which,  for  n  =  3/2,  gives  CQ/CO  =  —  6.  Thus,  with  an  infinitely  dis- 
tant object-point  a  biconvex  glass  Lens  has  the  least  Longitudinal 
Aberration,  viz.,  —  I5&V/I4'  when  the  curvature  of  its  first  surface 
is  six  times  as  great  as  the  curvature  of  its  farther  surface. 

271.     Case  of  a  System  of  Two  or  More  Thin  Lenses. 

If  the  optical  system  consists  of  a  system  of  m  Infinitely  Thin 
Lenses,  with  the  centres  of  their  surfaces  ranged  along  one  and  the 
same  straight  line,  we  can  determine  the  Longitudinal  Aberration  by 
means  of  the  formula  (284).  We  shall  employ  here  a  notation  entirely 
similar  to  that  used  above  in  the  case  of  a  single  Lens  (§  268)  ;  but  it 
should  be  noted  also  that  in  the  following  formulae  the  subscript  attached 
to  a  symbol  will  indicate,  not,  as  usually,  the  ordinal  number  of  the 
spherical  refracting  surface,  but  the  ordinal  number  of  the  Lens  to 
which  the  symbol  has  reference.  The  bundle  of  object-rays  is  sup- 
posed to  emanate  from  an  object-point  Ml  on  the  optical  axis,  and  the 
point  where  the  paraxial  image-rays  cross  the  axis  will  be  designated 
here  by  M*m,  and,  similarly,  the  point  where  the  outermost  ray  of  the 
bundle  of  image-rays  crosses  the  axis  will  be  designated  by  L'm.  For 
the  Longitudinal  Aberration  of  the  system  of  m  Lenses,  we  obtain: 


(297) 


§  272.J  Theory  of  Spherical  Aberrations.  393 

where 


In  this  formula  nk  denotes  the  relative  index  of  refraction  from  air 
into  the  medium  of  the  &th  Lens;  ck  and  <pk  denote  the  reciprocals 
of  the  radius  of  the  first  surface  and  the  primary  focal  length,  re- 
spectively, of  this  lens;  xk  denotes  the  reciprocal  of  the  intercept 
AkMk,  where  Mk  designates  the  point  where  paraxial  rays  cross  the 
axis  before  entering  the  &th  Lens;  hk  denotes  the  incidence-height  of 
the  outermost  ray  at  the  kih  Lens;  and,  finally,  um  =  AmM'm  is  the 
intercept  of  the  paraxial  image-rays. 

If  the  distances  that  separate  the  Lenses  are  all  negligible,  so  that 
we  have  a  System  of  m  Thin  Lenses  in  Contact,  the  formula  becomes  : 

u'  fa!  k=m 

M'mL'n  =  -  -~  £  <f>kZk.  (299) 

2       k=l 

Here  the  relation  xk+1  =  xk  +  <pk  will  also  be  of  service. 

272.  We  may  consider  somewhat  more  in  detail  the  special  case  of 
an  optical  system  consisting  of  Two  Infinitely  Thin  Lenses  in  Contact. 
The  condition  that  the  Longitudinal  Aberration  of  a  combination  of 
this  kind  shall  vanish  is: 


If  the  focal  lengths  of  the  two  Lenses,  or  their  reciprocals  <plt  <pv  are 
assigned,  and  if  also  we  know  the  reciprocal  xl  of  the  distance  uv  of 
the  axial  object-point  from  the  first  Lens,  then,  since  x2  =  x^  +  <plt 
the  analytical  condition  for  the  abolition  of  the  spherical  aberration 
will  be  an  equation  of  the  2nd  degree  in  cl  and  c2.  We  may,  therefore, 
choose  arbitrarily  the  value  of  one  of  these  two  magnitudes;  in  which 
case  there  will  always  be  two  values  of  the  other,  real  or  imaginary, 
which  will  fulfil  the  above  requirement. 

Since,  therefore,  we  have  here  two  arbitrary  variables  cl  and  c2 
and  only  one  equation  to  determine  them,  we  may  impose  one  other 
condition.  For  example,  a  very  natural  idea  would  be  to  make  the 
curvatures  of  the  second  surface  of  the  front  Lens  and  the  first 
surface  of  the  following  Lens  identical  (c{  =  c2),  so  that  the  two  Lenses 
could  be  cemented  together.  However,  if  the  two  Lenses  are  made  of 
different  kinds  of  glass,  with  unequal  co-efficients  of  dilatation,  a  com- 


394  Geometrical  Optics,  Chapter  XII.  [  §  273. 

bination  of  two  cemented  Lenses  might  become  distorted  under  the 
influence  of  changes  of  temperature. 

It  has,  therefore,  been  suggested  that  the  other  requirement  should 
be  the  so-called  HERSCHEL-  Condition;  that  is,  that  the  function 

<P1Z1  +  <p.2Z2 

should  vanish  not  only  for  the  particular  value  of  xl  but  also  for 
object-points  on  the  axis  very  near  to  the  point  Ml  to  which  the  value 
ocl  belongs.  This  condition  will  be  expressed  analytically)  by  the 
equation  : 

d 
0£  (<?&  +  <p2Z2)  =o; 

and,  thus,  we  obtain  a  second  equation  between  c1  and  c2,  as  follows: 


3»i  +  i    2      2(37*2  +  2)  37*2+1    2 

"  n   -  7  <Pl  ~      ~^i          W2  ~  7  --  7  ^2  =  °" 

/*•!  •!•  rl>2  7T2  —   1 

These  two  equations,  taken  simultaneously,  will  determine  completely 
the  forms  of  the  two  Lenses. 

If  now  we  impose  still  a  third  condition,  viz.,  that  the  combination 
of  the  Two  Thin  Lenses  in  Contact  shall  be  free  from  spherical  aberra- 
tion for  all  positions  of  the  object-point  on  the  axis,  so  that  the  funct- 
ion ^1Z1+^>2Z2  shall  vanish  for  all  values  of  xlt  then,  in  addition 
to  the  two  equations  above,  we  must  have  also: 

3*1  +  2          3^2  +  2. 

-<Pi  +  -        -<p2=o. 
HI  n.2 

Evidently,  in  order  to  satisfy  this  last  requirement,  ^  and  tpz  must 
have  opposite  signs;  that  is,  the  combination  must  consist  of  a  posi- 
tive Lens  and  a  negative  Lens.  Moreover,  with  the  actual  kinds  of 
glass  which  are  at  our  disposal  it  will  be  found  necessary  to  make  the 
curvatures  of  the  Lenses  exceedingly  great  in  order  to  comply  with  this 
last  requirement. 

ART.  84.     NUMERICAL   ILLUSTRATION    OF  METHOD    OF  USING   FORMULAE 
FOR  CALCULATION  OF  SPHERICAL  ABERRATION, 

273.  In  Chapter  X,  Art.  67,  we  computed  by  the  methods  of 
exact  trigonometrical  calculation  the  Longitudinal  Aberration  of  a 
large  Telescope  Object-Glass,  the  data  of  which  will  be  found  in  that 
place.  Merely  to  illustrate  the  use  of  the  formulae  which  we  have 


§  273.]  Theory  of  Spherical  Aberrations.  395 

obtained  here,  it  is  proposed  now  to  calculate  for  this  same  system  the 
First  Term  of  the  Spherical  Aberration  of  the  Edge-Ray  (§265),  and 
the  Lack  of  Detail  in  the  Image  (§  266).  The  formulae  employed  are 
the  following: 


2  k=m 


where 


Also,  for  the  Angular  Value  of  the  Indistinctness  or  Lack  of  Detail 
in  the  Image,  according  to  ABBE,  we  have  the  following  formula: 


Moreover,  in  order  to  find  the  value  of  hkfhl  for  each  surface,  we  have  : 


Thus,  for  the  first  surface  (&  =  i),  we  have:  h^jh^  —  i;  for  the  second 
surface  (k  =  2): 

for  the  third  surface  (k  =  3)  : 


and  for  the  fourth  (and  last)  surface  (k  =  4  =  m)  : 


Accordingly,  using  the  values  of  the  w's  as  found  in  Chapter  X,  Art. 
67,  we  obtain: 

Ig  w2  =  2.2430549  + 
clg  «;  =  7.7544706  + 
Ig  hjh,  =  9-9975255  + 


396 


Geometrical  Optics,  Chapter  XII. 


[§273. 


Ig  hjh,  =  9-9975255  + 
Ig  w3  =  1.8426804  + 

cig  «;  =  8.1572385  + 

Ig  hjh,  =  9.9974444  + 

lgwf  =  2.3518454  + 

clg  ti,  =  7.6462272  + 

Ig  *4/*l    =    9-9955I70  + 

The  following  scheme  exhibits  the  process  of  the  calculation : 


Clgtt* 

clg  n'k-i 
clg  n'k—\ 

clgu'k 
clgn'k 


c 

I/HkUk 


clgr* 
i/r. 

-I  /UK 
I/Tk  —  1/Uk 

Ig  (i/r*  —  I 
Ig  n*-i 

Ig/* 


IfCfc 

Igp* 


k  =  I 

k  =  2 

k  =  3 

k  =  4 

7.7544706+ 

9.8197020+ 

7-7569451  + 

9.8197020  + 

8.1573196+ 

o.ooooooo 

7.6481546+ 
9.7926080+ 

7.5766471  + 

8.1573196+ 

7.4407626+ 

8.1572385  + 

o.ooooooo 

7.6462272+ 
9.7926080+ 

7.7541648+ 

o.ooooooo 

7.5741726+ 

8.1572385+ 

7.4388352+ 

7.7541648+ 

+0.0037512 
o.ooooooo 

+0.0143628 
—0.0037727 

+0.0027469 

—0.0143655 

+0.0056776 
—  0.0027591 

+0.0037512 

+0.0105901 

—  0.0116186 

+0.0029185 

8.2232988+ 

8.0450343  — 

8.0721166  — 

7.3872161  + 

8.2232988+ 

0.0000000 

—  0.0110926 

+0.0057141 

—  0.0118064 

+0.0143655 

+0.0024390 
+0.0044479 

—0.0168067 

—  0.0261719 

—  0.0020089 

8.2254825  — 
0.1802980+ 

8.4178352- 

0.0000000 

7.3029583  — 

0.2073920  + 

8.2232988+ 

8.4057805  — 

8.4178352  — 

7.5103503  — 

6.4465976+ 
7.5741726+ 

o.ooooooo 

6.8115610+ 
8.0249001  + 
9.9901020+ 

6.8356704+ 
8.0651538  — 
9.9897776+ 

5.0207006+ 

7.4651597+ 
9.9820680+ 

4.0207702  + 

4.8265631  + 

4.8906018  — 

2.4679283  + 

P^=  +  1  04.  899-  1  o 

p*  =  4-  670.754  -iQ 

P4=   +        2.937  -IQ 


+  778.590-  io 


-8 


clg  2»i 


2P   =  +      1.266-iQ-8 


=  2.1024337  + 
=  0.0089660  + 
=  1.5563025  + 
=  4.4916704  + 
=  9.6989700  + 
7.8583426  + 


Accordingly,  we  find  : 


=  —  0.0072  inches.1 


1  TAYLOR,  computing  the  Spherical  Aberration  by  a  formula  equivalent  to  the  one 
employed  by  us,  obtains  a  different  value  and  one  which  agrees  very  closely  with  the 
exact  value.  But  there  appears  to  be  a  numerical  error  in  his  calculation  of  what  he  calls 
the  "  first  parallel  plate  correction". 


§  274.]  Theory  of  Spherical  Aberrations.  397 

It  will  be  perceived  that  the  value  of  the  longitudinal  aberration 
M'±L'±  thus  obtained  is  in  fact  rather  more  than  twice  as  great  as  the 
exact  value  obtained  in  Art.  67  by  the  rigourous  process  of  trigonomet- 
ric computation,  and  at  first  sight  it  might  appear,  therefore,  that  the 
approximate  value  was  utterly  unreliable.  However,  the  two  values 
are  of  the  same  order  of  magnitude,  and  a  little  reflection  will  convince 
anyone  that  in  this  particular  example,  at  least,  we  have  no  right  to 
expect  an  agreement  between  the  two  values  beyond  the  second  place 
of  decimals.  At  least  one  of  the  values  of  d'k  is  very  nearly  equal  to 
5°,  and  if  we  bear  in  mind  that  when  we  use  the  formula  of  the  first 
approximation  we  are  neglecting  all  terms  involving  the  powers  of 
this  angle  above  the  second,  we  can  easily  see  that  the  agreement 
above  in  the  first  two  figures  to  the  right  of  the  decimal-point  is  all 
that  we  could  look  for  here. 

In  order  to  find  ABBE'S  measure  of  the  Angular  Value  of  the  Lack 
of  Detail  in  the  Image  on  account  of  the  Spherical  Aberration,  we 
proceed  as  follows: 

IgSPfc  =  2.1024337  + 
3  Igfci  =  2.3344538  + 

Clg  2Wj   =  9.6989700  + 

lge,,m  =  4-I3S857S  - 

This  angle  is  expressed  here  in  radians.  It  will  be  found  to  be  less 
than  o".3. 

ART.  85.     CONCERNING  THE  TERMS  OF  THE  HIGHER  ORDERS  IN  THE 
SERIES-DEVELOPMENT  OF  THE  LONGITUDINAL  ABERRATION. 

274.  The  formulae  derived  in  Art.  82  were  based  on  the  assumption 
that  we  could  put  v  —  u  =  a02;  thereby  in  the  series-development  of 
the  expression  for  the  Longitudinal  Aberration  neglecting  all  the  terms 
after  the  first.  So  long  as  the  slope-angle  6  is  relatively  small,  this 
procedure  is  fairly  justified,  and  even  though  the  formulae  thus  ob- 
tained cannot  claim  to  be  entirely  accurate,  they  will  often  enable  us 
to  compute  very  approximately  the  magnitude  of  the  Spherical  Aber- 
ration. Applied  to  optical  systems  of  relatively  narrow  aperture,  the 
formulae  will  be  found  to  be  extremely  serviceable  in  so  far  as  they 
exhibit  clearly  the  effect  that  will  be  produced  by  a  variation  of  any 
one  of  the  factors  (radii,  intervals,  etc.)  that  are  involved  in  the 
problem:  so  that  the  optical  designer,  instead  of  having  to  grope  his 
way  by  means  of  tedious  trial-calculations,  can  proceed  methodically 
to  make  such  alterations  as  he  sees  will  tend  to  diminish  the  Spherical 


398  Geometrical  Optics,  Chapter  XII.  [  §  275. 

Aberration.  Especially,  in  the  design  of  the  Objectives  of  Telescopes 
— a  problem  which  ever  since  the  time  of  GALILEO  has  engaged  the 
attention  of  some  of  the  greatest  mathematicians  of  the  world — 
these  approximate  formulae  have  proved  to  be  of  the  greatest  value. 
If  the  Longitudinal  Aberration  5u  is  developed  in  a  series  of  ascend- 
ing powers  of  one  of  the  variables  a,  0,  <p  or  h,  it  is  obvious  that  the 
greater  the  relative  magnitude  of  this  variable,  the  more  terms  of  the 
series  will  it  be  necessary  to  take  account  of.  Thus,  provided  the 
slope-angle  0  is  not  too  great,  it  may  suffice  to  take  account  of  only 
the  first  two  terms  of  the  development,  and  then  we  may  write : 

du  =  a62  +  bB\ 

The  development  of  the  formulae  for  the  co-efficients  a  and  b,  by 
ABBE'S  Method  of  Invariants,  is  given  by  KOENIG  and  VON  ROHR  in 
their  treatise  on  Die  Theorie  der  sphaerischen  Aberrationen.1  The  re- 
current formula  obtained  in  this  way  for  the  aberration-co-efficient  of 
the  second  term,  viz.  b'm,  is  not  too  complex  to  be  often  very  service- 
able in  the  practical  design  of  optical  instruments;  but  the  co-efficients 
of  the  succeeding  terms  of  the  series  lead  to  exceedingly  complicated 
algebraic  expressions,  and  are  not  usually  of  much  value  on  this  ac- 
count, especially  also  as  we  begin  to  encounter  well-nigh  insurmount- 
able numerical  difficulties  in  trying  to  evaluate  by  means  of  these 
expressions  the  radii  of  the  spherically  corrected  system.  In  case  it 
is  necessary  to  take  account  of  these  higher  terms,  the  only  satis- 
factory procedure  is  to  resort  to  the  laborious  method  of  trigonomet- 
rical calculation  of  the  ray-paths.  After  a  number  of  trials  it  is  nearly 
always  possible  by  suitable  alterations  of  the  radii,  thicknesses,  etc., 
to  contrive  so  that  some  selected  ray  shall  emerge  from  the  system  so 
as  to  cross  the  optical  axis  approximately  at  the  same  point  as  the 
paraxial  image-rays;  and  although  this  by  no  means  implies  that 
any  other  ray  of  the  same  meridian  section  will  also  intersect  the 
axis  at  this  point,  it  is  usually  a  first  step  in  the  direction  of  dimin- 
ishing the  Longitudinal  Aberration.  The  method  is  very  fully  ex- 
plained, with  a  great  number  of  actual  numerical  illustrations,  in 
STEINHEIL  &  VOIT'S  Handbuch  der  angewandten  Optik  (Leipzig,  1891). 
275.  The  Aberration  Curve.  If  the  Longitudinal  Aberration  5u  of 
a  ray  of  incidence-height  h  is  developed  in  a  series  of  ascending  powers 
of  h,  and  if  we  take  account  of  only  the  first  two  terms,  we  may  write : 

8u  =  ah2  +  bh4', 

1This  is  Chapter  V  of  VON  ROHR'S  Die  Theorie  der  optischen  Instrumente  (Berlin,  1904); 
see  pages  217-219  and  pages  235-239. 


§  275.] 


Theory  of  Spherical  Aberrations. 


399 


where  a  and  b  are  co-efficients  independent  of  the  variable  h.1  If  a 
and  b  both  vanish,  the  Longitudinal  Aberration  will  be  zero  for  all 
values  of  h,  and  in  such  a  case  (which  never  actually  occurs)  the  optical 
system  would  be  entirely  free  from  aberration  for  the  axial  object-point 
in  question. 

If  we  suppose  that  the  co-efficients  a,  b  have  opposite  signs,  we 
shall  find  that  the  above  equation  represents  a  curve,  of  the  general 


M 


FIG.  134. 
ABERRATION-CURVE  :  CASE  OF  UNDER-COR- 


FiG.  135. 
ABERRATION-CURVE:  CASE  OF  OVER-COR- 


form  shown  in  Figs.  134  and  135,  which  is  symmetrical  with  respect  to 
the  tf-axis,  and  which  is  tangent  to  the  h-axis  at  the  origin.  This 
curve  is  called  the  Aberration  Curve.  For  the  value  du  =  o,  we  obtain: 


h  =  g  =  =b  V-  ajb\ 

consequently,  the  ray  whose  incidence-height  is  equal  to  g  will  cross 
the  optical  axis  at  the  point  M  where  the  paraxial  rays  converge,  and 
the  system  is,  therefore,  said  to  be  spherically  corrected  for  this  ray. 
For  all  values  of  h  comprised  between  h  =  o  and  h  =  g,  the  sign  of 
du  remains  unchanged,  so  that  all  the  intermediate  rays  will  be  either 
spherically  under-corrected  (8u  <  o),  as  in  Fig.  134,  or  spherically 
over-corrected  (du  >  o),  as  in  Fig.  135. 

.Moreover,  we  have  a  maximum  (or  minimum)  value  of  the  Longi- 
tudinal Aberration  du  at  the  origin  and  also  at  the  points  whose  ordi- 
nates  are  : 


1  These  co-efficients  a  and  6  are,  of  course,  not  the  same  as  the  co-efficients  denoted 
by  these  same  letters  in  the  development  of  M  in  a  series  of  ascending  powers  of  0. 


400  Geometrical  Optics,  Chapter  XII.  [  §  276. 

The  absolute  value  of  the  Longitudinal  Aberration  will  be  greatest, 
therefore,  for  the  ray  whose  incidence-height  is  j  =  g/1/2,  and  this 
value  is  nearly  equal  to  —  az/^.b.  The  smaller  this  greatest  value  is, 
the  more  nearly  will  the  system  be  spherically  corrected. 

Without  knowing  the  values  of  the  co-efficients  a  and  b,  the  Aberra- 
tion Curve  can  be  plotted  by  calculating  by  the  trigonometrical  for- 
mulae the  values  of  u  corresponding  to  given  values  of  the  incidence- 
height  h,  and  by  practical  opticians  this  method  is  used  to  exhibit 
graphically  the  performance  in  respect  to  spherical  aberration  of  the 
optical  system  as  finally  completed. 

Concerning  the  Choice  of  a  Suitable  Aperture  for  the  Objective,  the 
question  arises,  Which  ray  of  the  bundle  shall  be  "corrected  "  so  as 
to  cross  the  optical  axis  at  the  point  where  the  paraxial  rays  converge? 
According  to  GAUSS, l  if  H  denotes  the  radius  of  the  aperture  of  the 
objective,  we  should  choose  for  this  purpose  the  ray  for  which 


h  =  g=  ff  1/6/5. 
The  value 

h  =  g  =  H 


has  also  been  recommended  as  a  suitable  value  of  the  incidence- 
height  of  the  corrected  ray;  in  this  case  the  working  part  of  the 
spherical  refracting  surface  will  be  divided  by  the  circle  of  radius  g 
into  two  equal  zones,  so  that  half  of  the  refracted  rays  will  be  under- 
corrected  and  half  will  be  over-corrected. 

III.     THE  SINE-CONDITION.     (OPTICAL  SYSTEMS  OF  WIDE  APERTURE  AND  SMALL  FIELD 

OF  VISION.) 

ART.  86.     DERIVATION  AND   MEANING   OF  THE   SINE-CONDITION. 

276.  We  have  seen  that  it  is  possible  to  design  an  optical  system 
of  centered  spherical  surfaces  which  for  a  pair  of  conjugate  axial  points 
is  free,  or  practically  free,  from  spherical  aberration;  so  that  to  a 
homocentric  bundle  of  object-rays  proceeding  from  a  point  M  on  the 
optical  axis  there  will  correspond  a  homocentric  bundle  of  image-rays 
with  its  vertex  at  the  GAUSsian  image-point  M'.  If  the  optical  sys- 
tem consists  of  a  single  spherical  refracting  surface,  it  will  be  recalled 
that  it  was  the  pair  of  so-called  Aplanatic  Points  Z,  Z'  that  were  thus 
characterized  by  the  property  that  to  an  incident  chief  ray  crossing 
the  axis  at  Z  at  any  angle  B  corresponded  a  refracted  ray  crossing  the 

1  See  GAUSS'S  Letter  to  BRANDES,  given  in  GEHLERS  Physik.  Woerterbuch  (Leipzig,  1831), 
Bd.  vi.,  I.  Abt.,  S.  437.  This  letter  is  quoted  at  length  in  CZAPSKI'S  Theorie  dcr  optischen 
Inslrumente  (Breslau,  1893),  p.  96. 


§  276.]  Theory  of  Spherical  Aberrations.  401 

axis  at  the  conjugate  point  Z'  (§207).  But  this  was  not  the  only 
characteristic  of  this  remarkable  pair  of  points,  for  we  found,  also 
(§211,  Note  3),  that  the  slope-angles  6,  0'  of  the  incident  and  re- 
fracted rays  were  connected  by  the  relation: 

ny  sin  0  =  n'y'  sin  6', 

where  y'jy  =  Y  denoted  the  Lateral  Magnification  of  the  imagery  by- 
means  of  paraxial  rays  with  respect  to  the  pair  of  conjugate  axial 
points  Z,  Z'.  If  the  relation  between  the  Object-Space  and  the  Image- 
Space  were  a  collinear  relation  (as  it  would  be  if  all  the  rays  concerned 
were  paraxial  rays),  the  slope-angles  0,  0'  would  be  connected  by  the 
Law  of  ROBERT  SMITH  (§194),  viz.: 

ny  tan  0  =  n'y'  tan  0'; 

but,  since  for  finite  values  of  0,  0'  these  two  equations  cannot  both 
be  true  at  the  same  time,  it  is  manifest  that  the  correspondence  by 
means  of  wide-angle  bundles  of  rays  between  the  Aplanatic  Points 
of  a  single  spherical  refracting  surface  is  not  the  same  kind  of  corre- 
spondence as  we  have  in  the  ideal  case  of  optical  imagery.  Here  is 
a  matter,  therefore,  that  requires  to  be  investigated. 

The  mere  fact  that  an  optical  system  has  been  so  contrived  that 
for  a  pair  of  conjugate  axial  points  M,  M'  the  Spherical  Aberration  is 
sensibly  negligible,  by  no  means  implies  also  that  the  system  will  be 
free  from  aberration  for  any  other  object-point,  for  example,  for  a 
point  Q  very  near  to  M .  If  the  aperture  of  the  system  is  so  narrow 
that  the  rays  which  are  concerned  in  producing  the  image  may  be 
regarded  as  altogether  paraxial  rays,  we  know  that  to  an  infinitely 
small  object-line  M  Q  perpendicular  to  the  optical  axis  at  M  there  will 
correspond,  point  by  point,  an  infinitely  small  image-line  M'Q'  per- 
pendicular to  the  optical  axis  at  M'\  but,  in  general,  if  the  incident 
rays  which  come  from  the  object-point  Q  constitute  a  wide-angle 
bundle  of  rays,  only  those  rays  which  proceed  very  close  to  the  axis 
will  emerge  from  the  system  so  as  to  meet  in  the  correspond  ing  GAUSS- 
ian  image-point  Q'.  Even  in  those  cases  where  the  Spherical  Aber- 
ration with  respect  to  the  axial  points  M,  M'  has  been  most  completely 
abolished ,  the  points  of  the  image  which  are  not  on  the  axis  will  appear 
so  blurred  and  indistinct  that  the  diameters  of  their  aberration-circles 
are  actually  comparable  in  magnitude  with  their  distances  from  the 
axis.  According  to  ABBE/  the  explanation  of  this  indistinctness  is  to 

1  E.  ABBE:  Ueber  die  Bedingungen  des  Aplanatismus  der  Linsensysteme:  Sitzungsber. 
der  Jenaischen  Gesellschaft  fur  Med.  «.   Naturw.,  1879,  129-142;   also,  Gesammelle  Ab- 
handlungen,  Bd.  I,  213-226. 
27 


402  Geometrical  Optics,  Chapter  XII.  [  §  277. 

be  found  in  the  fact  that  the  images  of  the  object-line  MQ  (Fig.  136) 
produced  by  the  different  zones  of  the  spherically  corrected  system 
have  different  magnifications;  and,  thus,  although  all  these  images 
will  lie  along  the  same  line  perpendicular  to  the  optical  axis  at  M't 
being  of  unequal  lengths,  they  will  overlap  each  other  and  produce 

therefore  a  confused  image. 
If  the  angular  aperture  of 
the  objective  is  pretty 

__       large,    the    differences    in 

\        these    magnification-ratios 
may  amount  to  as  much  as 

FlG.  136.  50  per  cent,  or  more  of  the 

SINE-CONDITION.  lateral   magnification    pro- 

duced by  the  central  or  pa- 

raxial  rays.  Evidently,  under  such  circumstances  there  will  be  no 
imagery  at  all  in  any  practical  sense.  The  problem  consists,  there- 
fore, in  finding  the  condition  that  the  magnifications  of  all  the  differ- 
ent zones  of  the  objective  shall  be  equal  to  each  other,  that  is,  equal 
to  the  magnification 

Y  =  M'Q'/MQ 

of  the  imagery  by  means  of  paraxial  rays. 

277.  Consider  an  object-ray  u  proceeding  from  the  axial  object- 
point  M  to  which  corresponds  an  image-ray  u'  crossing  the  optical 
axis  at  the  point  Mr  conjugate  to  M\  and  let  0,  6'  denote  the  slope- 
angles  of  this  pair  of  corresponding  rays.  Since  the  optical  system 
is  supposed  to  be  spherically  corrected  with  respect  to  the  points  M, 
M',  to  an  infinitely  narrow  bundle  of  object-rays  whose  chief  ray 
is  u  will  correspond  an  infinitely  narrow  homocentric  bundle  of 
image-rays  whose  chief  ray  is  u'\  so  that  the  I.  and  II.  image- 
points  coincide  with  each  other  at  the  axial  point  M' .  We  saw  (§  245) 
that  within  the  infinitely  narrow  region  of  space  surrounding  the 
"mean"  incident  chief  ray  in  the  object-space  and  the  corresponding 
emergent  chief  ray  in  the  image-space,  there  was  a  collinear  corre- 
spondence between  the  plane-fields  TT,  TT'  of  the  Meridian  Rays  and 
also  between  the  plane-fields  TT,  TT'  of  the  Sagittal  Rays;  of  such  a 
character  that  to  an  infinitely  small  object-line  M V  lying  in  the  plane 
of  the  meridian  section  and  perpendicular  at  M  to  the  "mean"  inci- 
dent chief  ray  u  there  corresponds  an  infinitely  small  image-line  M'V1 
in  the  same  plane  and  perpendicular  at  M'  to  the  emergent  chief  ray 
u'\  and,  similarly,  to  an  infinitely  small  object-line  M W  lying  in  the 


§  277.]  Theory  of  Spherical  Aberrations.  403 

plane  TT  of  the  pencil  of  Sagittal  object-rays  and  perpendicular  at  M 
to  the  "mean"  incident  chief  ray  u  there  corresponds  an  infinitely 
small  image-line  M'W  in  the  plane  TT'  of  the  pencil  of  Sagittal  image- 
rays  and  perpendicular  at  M'  to  the  chief  image-ray  u' . 
We  shall  use  the  symbols 

7w  =  M'V'/MV,     Fu  =  M'W'IMW, 

to  denote  the  lateral  magnifications  of  the  Meridian  and  Sagittal  Rays, 
respectively.  The  line-elements  MW  and  M'W  are  perpendicular  to 
the  optical  axis  at  M  and  M',  respectively;  but  the  same  thing  is  not 
true  with  respect  to  the  line-elements  MV  and  M V .  If  in  the 
meridian  plane  we  draw  VR,  V'R'  perpendicular  at  F,  V  to  MV, 
M'V  and  meeting  in  R,  R'  the  axis-ordinates  erected  at  M,  M', 
respectively,  so  that 


then 


MV  =  MR-COS  0,     M'V  =  M'R'-cos  0'; 


M'R'          cos0 
MR  =     Ucos0'; 


and  in  order  that  the  image  at  M'  of  a  plane  element  perpendicular  to 
the  optical  axis  at  M  shall  be  identical  with  the  GAUSsian  image,  or 
the  image  produced  by  means  of  the  central  (paraxial)  rays,  we  must 
have: 

M'R'          '  M'' 


MR       ~MW        MQ 
that  is, 

cos0 


for  all  values  of  the  slope-angle  0. 
If 


d\ 


denote  the  angular  magnifications,  or  "convergence-ratios",  of  the 
incident  and  emergent  pencils  of  Meridian  and  Sagittal  Rays,  respect- 
ively, then,  since  the  formulae  which  were  deduced  in  the  case  of 
Collinear  Imagery  are  applicable  here,  we  have  (see  Chap.  VII,  §  179, 
and  Chap  XI,  §  246)  the  following  relations: 

F-Z,  =  Y-Z,  =n/n', 


404  Geometrical  Optics,  Chapter  XII.  [  §  277. 

where  n  and  n'  denote  the  refractive  indices  of  the  media  of  the  inci- 
dent and  emergent  rays,  respectively. 

Let  us  consider,  first,  the  Imagery  in  the  Plane  of  the  Meridian 
Section  of  the  infinitely  narrow  bundle  of  incident  rays  whose  chief 
ray  is  u.  Obviously, 

_dtf_ 
Z»-  dd' 

and  from  the  above  relations  we  obtain: 

M'R'  _    n  cos  6  dB    _  rn-d(sin8) 
MR   ~  n'  cos  tf  d6'  "  n'  •  d  (sin  0')' 

This  equation  shows  that  the  lateral  magnification  perpendicular  to 
the  optical  axis  at  the  points  M,  M'  produced  by  the  Meridian  Rays 
depends  on  the  slope-angle  6  of  the  chief  incident  ray  u\  and,  hence, 
the  condition  that  this  magnification  shall  have  the  same  value  for 
all  values  of  the  slope-angle  0,  between  the  value  6  =  o  and  the  value 
of  B  for  the  edge-ray  is: 

n  -  d  (sin  0)   _ 

i>~\   ==  •*  » 


n'-d  (sin  0') 

and,  since  this  equation  must  be   satisfied  by  all  values  of  6,  0',  in- 
cluding very  small  values,  it  may  be  written: 

sin  0       n'  .      . 

;  :     s?  •***;;  (300) 

In  the  next  place,  we  proceed  to  consider  the  Imagery  of  the  Sagittal 
Rays  of  the  same  infinitely  narrow  bundle  of  rays.  The  value  of  the 
angular  magnification  in  the  Sagittal  Section  may  easily  be  found  by 
imagining  the  figure  to  be  rotated  about  the  optical  axis  through  a 
very  small  angle,  in  which  case  the  angles  between  the  initial  and 
final  positions  of  the  chief  incident  and  emergent  rays  u,  u'  will  be  the 
angles  d\,  d\'  whose  ratio  d\'/d\  is  equal  to  Zu.  According  to  formula 
(251)  of  Chap.  XI  and  formula  (185)  of  Chap.  IX,  we  have  for  the  &th 
spherical  surface: 


and,  since 


k=l 


§  278.]  Theory  of  Spherical  Aberrations.  405 

we  shall  find : 

—  _  sin  6'm  _  sin  0' 
"""sin^  =  sin  6  ' 

since  here  we  write  0  and  0'  in  place  of  0l  and  0'm,  respectively. 

The  lateral  magnification  Fu  of  the  the  Imagery  by  means  of  the 
Sagittal  Rays  must  be  equal  to  the  lateral  magnification  Y  of  the 
imagery  by  means  of  the  Paraxial  Rays;  and,  hence,  since 

T.-Z.  =  F-ZU  =  n/n', 
we  obtain  here  also: 

sin  0       n' 

~       ~r\l    =  * 

sin  0       n 

as  the  condition  that  the  magnification  of  the  Imagery  by  means  of 
the  Sagittal  Rays  shall  be  constant  and  equal  to  that  by  means  of  the 
Paraxial  Rays;  and  this  condition  is  seen  to  be  precisely  the  same  as 
was  found  above  for  the  Imagery  by  means  of  the  Meridian  Rays. 
It  will  be  observed  also  that  it  is  likewise  identical  with  the  character- 
istic relation  which  we  found  to  be  true  always  in  regard  to  the  pair 
of  aplanatic  points  of  a  single  spherical  refracting  surface  (§  211, 
Note  3). 

The  law  here  derived,  known  as  the  Sine-Condition,  is  one  of  the 
most  important  of  the  valuable  contributions  of  ABBE1  to  the  theory 
of  Optical  Instruments.  It  may  be  stated  as  follows: 

The  necessary  and  sufficient  condition  that  all  the  zones  of  the  spheri- 
cally corrected  optical  system  shall  produce  equal-sized  images  at  the 
axis-point  M' ,  conjugate  to  the  axial  object-point  M,  is  that,  for  all  rays 
traversing  the  system,  the  ratio  of  the  sines  of  the  slope-angles  of  each  pair 
of  corresponding  incident  and  emergent  rays  shall  be  constant;  that  is, 

sin  0/sin  0'  =  constant. 

The  value  of  this  constant,  as  we  see  from  formula  (300),  is  n'Yjn. 

278.  Other  Proofs  of  the  Sine-Law.  The  so-called  Sine-Condition 
as  enunciated  by  ABBE,  in  1873,  for  the  special  case  of  a  centered 
system  of  spherical  refracting  surfaces  might  have  been  seen  to  be 

1  E.  ABBE:  Beitraege  zur  Theorie  des  Mikroskops  und  der  mikroskopischen  Wahr- 
nehmung:  M.  SCHULTZES  Archiv  fiir  mikroskopische  Anatomie,  IX  (1873),  413-468.  Also, 
Gesammelte  Abhandlungen,  Bd.  I,  45-100.  See  also  paper  entitled:  Ueber  die  Bedingungen 
des  Aplanatismus  der  Linsensysteme:  Silzungsber.  der  Jenaischen  Gesellschaft  fur  Med.  u. 
Naturw.,  1879,  129-142;  reprinted  in  CARLS  Repertorium  der  Exper.-Phys.,  XVI  (1880), 
303-316,  and  in  Gesammelte  Abhandlungen,  Bd.  I,  213-226. 


406  Geometrical  Optics,  Chapter  XII.  [  §  278. 

contained  in  a  far  more  general  law  of  CLAUSius's1  based  on  the 
Second  Fundamental  Principle  of  Thermodynamics;  which  may  be 
stated  thus: 

If  the  energy  radiated  by  an  element  of  surface  da,  in  a  medium  of 
refractive  index  n,  by  a  bundle  of  rays  of  solid  angle  dw,  is  transmitted 
entirely  to  an  element  of  surface  da' ,  in  a  medium  of  refractive  index 
n'y  by  a  bundle  of  rays  of  solid  angle  dco',  then  we  must  have  the  fol- 
lowing equation: 

n2  •  cos  6  •  du        da' 
n'2- cosS' -da'  =  da  ' 

where  6,  0'  denote  the  angles  between  the  chief  rays  and  the  corre- 
sponding surface-normals. 

Applied  to  the  case  of  an  optical  system  of  centered  spherical  surf- 
aces, to  the  axis  of  which  the  surface-elements  da,  da'  are  supposed 
to  be  perpendicular,  this  equation  is  easily  reducible  to  the  form  given 
by  formula  (300).  For  in  this  special  case  the  magnitudes  0,  8' 
evidently  denote  the  slope-angles  of  the  incident  and  refracted  rays, 
and 

dco        sin  6  •  dd 

cfa'=  sm0'-d0" 

so  that  CLAUSIUS'S  equation  becomes: 

n2-d(sin20)   ^da' 
n'2'd(sm20')  ~  da  J 

and,  since  da' /da  —  Y2,  we  obtain  by  integration: 

sin  0       n' 


sin  0'      n 


Y. 


Applying  the  Law  of  the  Conservation  of  Energy  to  the  Radiation 
of  Light,  HELMHOLTZ2  has  given  also  another  mode  of  deducing  ABBE'S 
Sine-Condition,  which  is  interesting,  inasmuch  as  this  important  re- 
sult is  thus  obtained  from  still  another  point  of  view. 

Finally,  let  us  mention  here  the  extremely  simple  and  elegant  proof 

1  See  BROWNE'S  English  Translation  of  CLAUSIUS'S  Mechanical  Theory  of  Heat  (London, 
1879),  p.  321.     The  law  of  CLAUSIUS'S  here  referred  to  was  first  published  in  the  cele- 
brated paper,  Die  Concentration  von  Waerme  und  Lichtstrahlen  und  die  Grenzen  ihrer 
Wirkung:  POGG.  Ann.,  cxxi.  (1864),  S.  i. 

2  H.  HELMHOLTZ  :    Die  theoretische  Grenze  fur  die   Leistungsfaehigkeit  der  Mikro- 
skope:  POGG.  Ann.,  Jubelband,  1874,  557-584.     See  also  Wissenschaftliche  Abhandlungen, 
II,  p.  185. 


§  279.]  Theory  of  Spherical  Aberrations.  407 

of  the  Sine-Law  published  by  Mr.  HOCKIN/  which  is  based  on  the 
general  law  of  the  equality  of  the  optical  lengths  (§  38)  of  all  the  ray- 
paths  between  the  pair  of  conjugate  axial  points  M,  M'  for  which  the 
system  is  assumed  to  be  spherically  corrected. 

ART.  87.     APLANATISM. 

279.  We  must  explain  here  the  meaning  that  is  to  be  attached 
to  the  term  "aplanatic" ,  as  it  is  employed  by  ABBE  and  modern 
writers  on  Optics.  Formerly,  this  word  was  applied  to  an  optical 
system  merely  to  mean  that  it  was  free  from  spherical  aberration, 
and  this  is  the  sense  in  which  the  term  is  used  by  CODDINGTON, 
HERSCHEL,  etc.  But,  according  to  ABBE,  in  order  for  an  optical 
system  to  be  aplanatic,  it  must  fulfil  each  of  two  requirements, 
viz.:  (i)  It  must  be  free  from  spherical  aberration  for  a  pair  of 
conjugate  axis-points  M,  M'',  and  (2)  The  sine-condition  must  also 
be  satisfied  for  this  pair  of  points  M,  M'.  Thus,  the  aplanatic  pair 
of  points  Z,  Z'  of  a  spherical  refracting  surface  are  rightly  so- 
called,  because  not  merely  are  these  points  free  from  aberration, 
but,  as  we  have  seen,  they  fulfil  the  Sine-Condition  also.  On  the 
other  hand,  the  focal  points  of  a  reflecting  ellipsoidal  surface  are 
not  aplanatic,  because  they  do  not  satisfy  the  Sine-Condition,  and  the 
same  observation  applies  also  with  respect  to  the  infinitely  distant 
axial  point  and  the  focal  point  of  a  parabolic  reflector. 

Accordingly,  the  Aplanatic  Points  of  an  optical  system  are  the  points 
en  the  axis  for  which  the  spherical  aberration  is  abolished,  and  which 
at  the  same  time  satisfy  the  Sine-  Condition. 

ABBE2  has  described  a  very  ingenious  and  simple  mode  of  testing 
the  aplanatism  of  a  lens-system;  consisting  in  viewing  through  the 
system  a  certain  sheaf  of  concentric  hyperbolae,  the  plane  of  the  object- 
iigure  being  placed  perpendicularly  to  the  axis  with  the  common  centre 
at  the  proper  distance  from  the  aplanatic  point;  which  should  yield  as 
image  two  sheaves  of  mutually  perpendicular,  equidistant  parallel  lines 
(see  §  291).  By  means  of  this  device,  ABBE  has  investigated  the  older 
typesof  microscopes,  and  he  has  shown  that,  long  before  the  publication, 

1  CHARLES  HOCKIN:  On  the  estimation  of  aperture  in  the  microscope:  Journ.  Royal 
Mic.  Soc.,  (2),  IV  (18847,  337-346.     See  also  J.  D.  EVERETT'S  note  on  HOCKIN'S  proof  of 
the  Sine  Condition,  Phil.  Mag.,  (6),  IV  (1902),  p.  170.     HOCKIN'S  Proof  of  the  Sine-Con- 
dition will  be  found  given  also  in  the  pth  edition  of  MUELLER-POUILLET'S  Lehrbuch  der 
Physik.  Bd.  II,  Optik,  and  in  DRUDE'S  Lehrbuch  der  Oplik. 

2  E.  ABBE:  Ueber  die  Bedingungen  des  Aplanatismus  der  Linsertsysteme:  Sitzungsber. 
der  Jenaischen  Gesellschaft   fur  Med.  u.   Naturw.,  1879,  129-142;   also,  Gesammelte  Ab- 
handlungen,  I,  213-226;  also,  reprinted   in  CARLS   Rep.  der  Exper.-Phys.,  XVI  (1880), 
303-316. 


408  Geometrical  Optics,  Chapter  XII.  [  §  280. 

in  1873,  of  the  Sine-Condition,  microscope-designers,  without  knowing 
it,  had  all  more  or  less  perfectly  fulfilled  this  essential  requirement 
along  with  the  abolition  of  the  spherical  aberration.  As  LuMMER1 
observes,  this  is  only  another  of  the  many  instances  in  which  correct 
practice  has  preceded  theory. 

ART.  88.     THE  SINE-CONDITION  IN  THE  FOCAL  PLANES. 

280.  It  has  been  pointed  out  (§  276)  that  the  imagery  which  we 
obtain  when  the  Sine-Condition  is  fulfilled  ^is  not  governed  by  the  same 
laws  as  we  have  in  the  case  of  Collinear  Imagery.  This  difference  is 
made  strikingly  manifest,  for  example,  if  the  aplanatic  pair  of  points  are 
the  infinitely  distant  point  of  the  optical  axis  and  one  of  the  Focal  Points 
of  the  optical  system;  as  we  shall  proceed  to  show.  Let  MB  be  an 
incident  ray  proceeding  from  the  axial  point  M  and  meeting  the  first 
surface  of  the  system  at  B,  and  let  us  put: 


where  A  designates  the  vertex  of  the  first  spherical  surface.  If  k 
denotes  the  incidence-height  of  this  ray  at  this  surface,  then 

h 

sm  6  =  —  y  . 

Moreover,  if  x  =  FM  denotes  the  abscissa  of  the  object-point  M  with 
respect  to  the  Primary  Focal  Point  F,  then  (see  Chap.  VII,  §  179) 
the  lateral  magnification  of  the  imagery  by  means  of  paraxial  rays  is  : 


where  /  denotes  the  Primary  Focal  Length  of  the  optical  system.     And 
since  (§  193) 


where  er  denotes  the  Secondary  Focal  Length  of  the  system,  evidently, 
we  may  write  : 

V  -    -~e- 
n'x' 

If  M,  M'  are  the  pair  of  aplanatic  points  of  the  system,  the  Sine-Condi- 

1  See  MUELLER-POUILLET'S  Lehrbuck  der  Physik,  Bd.  II,  Optik,  neunte  Auflage,  Art. 
191. 


§  281.]  Theory  of  Spherical  Aberrations.  409 

tion  expressed  by  formula  (300)  may  be  put  in  the  following  form: 


_h W_ 

sin  0'       x 

And  if  we  suppose  now  that  the  object-point  M  is  the  infinitely  distant 
point  E  of  the  optical  axis,  and,  consequently,  the  image-point  M' 
coincides  with  the  secondary  focal  point  E',  then  /  =  x  =  oo,  in  which 
case  we  find : 


sin0' 

Similarly,  for  the  case  of  an  infinitely  distant  image-point  F'  cor- 
responding to  an  object-point  at  the  Primary  Focal  Point  F,  we  should 
obtain : 

—  =/ 

sin0 

If,  therefore,  supposing  that  the  aplanatic  pair  of  points  is  the  pair 
Ej  E',  to  which  the  first  of  these  two  equations  applies,  we  describe 
around  the  Secondary  Focal  Point  E'  as  centre  a  sphere  of  radius  ef, 
all  the  points  of  intersection  of  the  parallel  object-rays  with  their  cor- 
responding image-rays  will  lie  on  the  surface  of  this  sphere,  whereas 
in  the  case  of  Collinear  Imagery,  these  points  of  intersection  of  the 
incident  and  emergent  rays  all  lie  in  the  Secondary  Principal  Plane 
which  touches  the  above-mentioned  sphere  at  its  vertex. 

ART.  89.     ONLY   ONE  PAIR  OF  APLANATIC  POINTS  POSSIBLE. 

281.  When  an  optical  system  is  so  contrived  that  for  a  certain 
pair  of  points  M ,  M'  on  the  optical  axis  not  only  is  the  spherical  aber- 
ration abolished  but  at  the  same  time  the  Sine-Condition  is  fulfilled, 
a  flat  element  of  luminous  surface  placed  normally  to  the  axis  at  M 
will  be  distinctly  delineated  as  a  flat  surface-element  at  Mf  by  bundles 
of  rays  of  any  angular  width  (not  exceeding  the  angular  aperture  of 
the  system) :  but  it  by  no  means  follows  that  the  system  will  give  at 
Mf  a  distinct  image  of  a  plane  area  at  M  of  finite  dimensions;  nor, 
indeed,  that  it  will  produce  such  an  image  even  of  an  element  of  surface 
if,  it  is  situated  at  any  other  place  on  the  axis.  In  fact,  an  optical 
system  cannot  have  even  two  pairs  of  adjacent  aplanatic  points;  for 
if  this  were  possible,  the  system  would  have  to  be  spherically  corrected 
for  both  pairs  of  points,  and  this  requirement,  as  we  shall  show,  is  in- 
compatible with  the  condition  that  either  of  the  two  pairs  of  points 
is  aplanatic. 


410  Geometrical  Optics,  Chapter  XII.  [  §  281. 

In  the  diagram  (Fig.  137)  M,  M'  are  supposed  to  be  a  pair  of  apla- 
natic  points  of  the  optical  system.  A  ray  MB  emanating  from  the 
object-point  M  and  inclined  to  the  axis  at  an  angle  6  will,  after  trav- 
ersing the  system,  emerge  so  as  to  cross  the  axis  at  the  image-point 
M',  the  slope  of  the  image-ray  being  denoted  by  6'.  This  pair  of 
corresponding  rays  may  be  regarded  as  the  chief  rays  of  two  infinitely 


FIG.  137. 
AN  OPTICAL  SYSTEM  CAN  HAVE  ONLY  ONE  PAIR  OF  APLANATIC  POINTS  M,  M'  . 

narrow  pencils  of  corresponding  Meridian  Rays;  let  I'  designate  the 
position  on  the  emergent  chief  ray  of  the  Secondary  Focal  Point  of 
this  pencil  of  Meridian  Rays  (see  §§235,  246).  The  image  M'Rr  of 
an  infinitely  small  object-line  MR  perpendicular  to  the  optical  axis  at 
M  will  be  determined  by  constructing  the  path  through  the  system  of 
a  ray  proceeding  from  R  parallel  to  MB  which  will  emerge  in  a  direct- 
ion very  nearly  the  same  as  that  of  the  emergent  chief  ray,  and  which 
will  intersect  this  ray  at  /',  and  which,  by  its  intersection  with  the 
normal  to  the  optical  axis  at  Mr  will  determine  the  image-point  R' 
corresponding  to  R.  Let  P,  Pf  designate  the  points  where  this  ray 
crosses  the  axis  before  and  after  refraction  through  the  optical  system. 
The  pair  of  axial  points  P,  P'  are  adjacent  to  the  aplanatic  pair  of 
points  My  M'\  and,  therefore,  let  us  write: 

MP  =  dx,     M'P'  =  dx'. 

Let  us  now  assume  also  that  the  optical  system  is  spherically  corrected 
for  the  points  P,  P',  so  that  they  also  are  a  pair  of  conjugate  points; 
in  which  case  the  ratio  dx'  /dx  will  be  the  value  of  the  axial  magnifica- 
tion, at  the  points  M,  M',  of  the  imagery  by  means  of  paraxial  rays. 
Hence  (see  Chap.  VII,  §  179),  we  find: 


dx       n      ' 

where  Y  denotes  the  lateral  magnification,  at  the  conjugate  points 
M,  M',  of  the  imagery  by  means  of  paraxial  rays. 


§281.]  Theory  of  Spherical  Aberrations.  411 

Now  from  the  figure  we  obtain: 

MR  =  -  dx-tanB,     M'R'  =  -  e 


since    /.M'P'R'  differs  from  the  angle  6'  by  only  an  infinitesimal 
magnitude;  and,  hence, 

dxf      M'R'  tan  0 

dx  ==  MR  tan0'* 

Here,  we  may  recall  that  in  §  277  we  found  : 
M'R'        n-cos  B'dB 


and  therefore  equating  the  two  expressions  above  for  dx'/dx,  and  at 
the  same  time  introducing  this  last  relation,  we  obtain  the  following 

equation  : 

/2 

sin0-</0  =  -2  Y^-smB'-dd'', 
which,  being  integrated,  gives: 


where   C  denotes  the  integration-constant.     The  value  of  C  can  be 
found  by  putting  B  =  Bf  =  o;  thus,  we  obtain: 


n 
Substituting  this  value  of  C  in  the  above  result,  we  find  : 

n'2 
i  -  cos  B  =  —2  F2(i  -  cos  0'), 

tlr 

which  can  be  written  finally  as  follows: 

0 
sin-         . 


sin- 

Evidently,  this  equation  cannot  be  satisfied  at  the  same  time  with  the 
Sine-Condition  expressed  by  equation  (300)  .  Consequently,  an  optical 
system  can  have  only  one  pair  of  aplanatic  points. 

This  result  might  have  been  established  immediately  by  merely  re- 
marking again  (§  276)  that  the  Sine-Condition  Imagery  is  essentially 


412  Geometrical  Optics,  Chapter  XII.  [  §  283. 

different  from  Collinear  Imagery.  Now  we  know  that  if  as  many  as  two 
elements  of  surface  perpendicular  to  the  axis  are  portrayed  by  similar 
surface-elements  also  perpendicular  to  the  optical  axis,  the  Imagery 
must  be  Collinear,  and  hence  it  follows  that  tha  Sine-Condition  cannot 
be  satisfied  for  two  pairs  of  axial  points.  Thus,  for  example,  the  ob- 
jective of  a  microscope  must  always  be  computed  for  that  pair  of 
aplanatic  points  for  which  it  is  to  be  used;  and  in  order  to  obtain  a 
distinct  image  of  the  object,  the  latter  must  be  placed  at  the  aplanatic 
point  of  the  Object-Space. 

It  appears,  therefore,  that  with  all  the  means  at  his  disposal,  the 
utmost  that  the  practical  optician,  employing  wide-angle  bundles  of 
rays,  can  hope  to  achieve  is  the  approximate  realization  of  one  or  other 
of  two  theoretical  possibilities:  To  produce  a  perfectly  sharp  image 
(i)  Either  of  an  indefinitely  small  element  of  surface  perpendicular  to 
the  axis,  (2)  Or,  else,  of  an  indefinitely  small  element  of  the  axis  itself. 
It  is  practically  impossible  to  obtain  a  sharp  image  of  even  an  indefi- 
nitely small  axial  element  of  volume;  for  the  conditions  which  are 
required  to  be  fulfilled  in  order  to  portray  distinctly  its  dimension 
parallel  to  the  optical  axis  are  at  variance  with  the  conditions  that 
must  be  satisfied  in  order  to  produce  a  distinct  image  of  its  lateral 
dimensions. 

ART.   90.     DEVELOPMENT   OF  THE   FORMULA  FOR   THE   SINE-CONDITION 

ON  THE  ASSUMPTION  THAT  THE   SLOPE-ANGLES 

ARE    COMPARATIVELY   SMALL. 

282.  Let  us  assume  now  that  the  effective  bundles  of  rays  are 
limited  by  a  suitable  stop  so  that  the  slope-angles  0  are  all  compara- 
tively small  —  so  small  that  we  may  neglect  powers  of  6  above  the 
third.  The  following  method  of  development  is  practically  the  same 
as  that  given  by  KOENIG  and  VON  RoHR.1 

Since,  according  to  formula  (185), 

sin  O'k    _  lk 
sin  0fc_!  ~  4  * 
we  have,  evidently: 


«;  sine!.  _  nl.  g  ^in  £_  =  <  g  J. 

Wj  sin  0!       n±  k=._i  sin  0^       nv  A=i  l'k 


283.     Let  us  first  obtain  the  development  of  the  ray-length  in  a 
series  of  ascending  powers  of  the  central  angle  <p.     Since,  by  the  second 

1  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen  :  Chapter 
V  of  VON  ROHR'S  Die  Theorie  der  optischen  Instrumente  (Berlin,  1904),  Bd.  I,  302-304. 


§  284.]  ,     Theory  of  Spherical  Aberrations.  413 

of  formulae  (180), 

v  —  r  =  /  •  cos  0  —  r  •  cos  <pt 

we  obtain,  neglecting  powers  of  6  and  <p  above  the  third, 


Now 


i 

l\  i  --  I  =  v  —  r—  , 

2  2 


(         **\(    *f\        (    ^      r*\ 
=  [v  —  r—  )(  i  +  —  ]  =  v[i  +  ----  —). 

V  a/V      •-»/         \        2       U2  J 


u2  ' 


and,  hence,  finally,  we  obtain: 

C30D 


where  J  denotes  the  so-called  zero-invariant  (§  126). 
Since 

v  =  u  +  8u,     r2<p2  =  h2, 

we  may  also  write  this  formula  as  follows: 

(302) 


And  for  the  ray-length  /'  of  the  refracted  ray  we  have  merely  to  prime 
the  letters  n  and  u  in  this  formula. 

To  the  same  degree  of  approximation,  we  obtain,  therefore,  for  the 
ratio  1  1  1'  the  following  formula: 

/        u  (        ,9/i  du\ 

77  =  -7(i  +  ^2-A-  --  A-  ).  (303) 

/'      u'  \  2     nu          u  J 

284.     Thus,  re-introducing  the  subscripts,  we  obtain  : 


nt    sin  0l 


gtyr  j  i  ^-'gAf^lff** 

*"  *      \  **  /a     S     \  «  AJ  iA  <  ' 


If  this  expression  is  to  be  constant  for  all  values  of  0t  (or  &,),  then 
we  must  have  : 


414  Geometrical  Optics,  Chapter  XII.  [  §  284. 

Now,  since  (see  formulae  (270)) 

J  -  J  =  n  (  -  -  -  }  =  n '  (  -,  -  -t  Y 
\u      uj          \u'      u  ) 

we  have: 

A  du          i      (  n-8u       A  n-du\ 

A  —  =  -z -.  I  A A  — 2"  J, 

u       J  -  J\      u  u2  J' 

and 


u  —  u 
and,  hence, 


But,  since 

we  have  evidently: 


since  in  the  present  case,  in  which  the  system  is  supposed  to  be  spheri 
cally  corrected  for  the  two  axial  points  Mv  M'm,  we  must  have: 

8Ul  =  o  =  dum. 
According  to  formula  (281  a),  we  have: 


n-du          -   2  2        i 
A  —  2~  =  —  \nJ  -A  —  ; 
w  nu 


and,  hence,  we  find: 


i    kJk-Jk 


k 
Accordingly,  the  Sine-Condition  may  be  expressed  as  follows: 


or 


Let  Qt  designate  the  end -point  of  the  infinitely  short  object-line 
MlQl  perpendicular  to  the  optical  axis  at  Mlt  and  let  /i,  denote  the 
incidence-height  of  the  paraxial  object-ray  which,  proceeding  from  Ql 


§  285.]  Theory  of  Spherical  Aberrations.  415 

is  directed  towards  the  centre  Ml  of  the  Entrance-Pupil  (§257),  and 
let  hk  denote  the  incidence-height  of  this  ray  at  the  kth  spherical  surf- 
ace. Introducing  the  relation  given  by  formula  (155)  of  Chapter 
VIII,  viz.: 

MftCJ*  -  Jk)  =  V»i(7i  -  -A), 

which,  in  the  way  it  is  employed  here,  is  admissible,  since  we  neglect 
magnitudes  above  the  third  order,  we  obtain  finally  the  formula  for 
the  Sine-Condition  in  the  following  form: 

,  \ 

=  o.  (304) 

SEIDEL1  notes  the  fact  that  FRAUNHOFER  in  his  characteristic  con- 
struction of  the  telescope-objective,  appears  to  have  satisfied  this 
condition,  and  he,  therefore,  calls  formula  (304)  the  FRAUNHOFER 
Condition. 

If  this  condition  is  fulfilled,  along  with  the  condition  of  the  aboli- 
tion of  the  spherical  aberration  for  the  conjugate  axial  points  Mlt  M'm> 
we  shall  have  (cf.  Chapter  VI,  §  138) : 

n^  sin  0        nn,  -F?  UT.       I 


n,  sin  6l       n,  £3  uk      Y  ' 

IV.     ORTHOSCOPY.     CONDITION  THAT  THE  IMAGE  SHALL  BE  FREE  FROM  DISTORTION. 

ART.  91.     DISTORTION  OF  THE  IMAGE  OF  AN  EXTENSIVE  OBJECT  FORMED 
BY  NARROW  BUNDLES  OF  RAYS. 

285.  In  case  the  object  to  be  depicted  is,  say,  a  plane  surface  of 
finite  dimensions  placed  perpendicular  to  the  optical  axis  of  the  Lens- 
System,  our  only  chance  of  obtaining  an  approximately  correct  image 
will  be  by  introducing  a  small  circular  stop,  or  diaphragm,  whose  duty 
will  be  to  limit  the  angular  widths  of  the  operative  bundles  of  rays 
emanating  from  the  various  points  of  the  object.  It  is  obvious  that 
this  mode  of  producing  an  image  will  be  attended  also  by  a  number  of 
difficulties  of  one  kind  and  another,  which  may  be  described  in  a 
general  way  as  aberrations  due  to  the  obliquity  of  the  rays  proceeding 
from  the  lateral  parts  of  the  object.  In  general,  a  plane  object  will 
not  be  reproduced  by  a  plane  image,  but  on  account  of  the  astigmatism 
of  the  narrow  bundles  of  rays,  the  image  will  be  resolved  into  a  double 
image,  symmetrically  situated  with  respect  to  the  optical  axis  on  two 

1  L.  SEIDEL:  Zur  Dioptrik.  Ueber  die  Entwicklung  der  Glieder  3ter.  Ordnung,  welche 
den  Weg  eines  ausserhalb  der  Ebene  der  Axe  gelegenen  Lichtstrahles  durch  ein  System 
brechenden  Medien,  bestimmen:  Astr.  Nach.,  No.  1029,  xliii.  (1856).  See  Section  9  of 
SEIDEL'S  paper. 


416  Geometrical  Optics,  Chapter  XII.  [  §  286. 

curved  surfaces,  called  the  "astigmatic  image-surfaces"  (§  295). 
However,  passing  over  for  the  present  both  of  these  difficulties,  and 
assuming  that  the  aberrations  which  produce  astigmatism  and  curva- 
ture of  the  image  have  been  eliminated  to  some  extent  at  least,  so  that 
we  have  a  fairly  sharp,  flat  image,  even  under  these  conditions  we  may 
still  find  that  the  image  does  not  reproduce  the  object  faithfully,  but  is 
distorted.  This  latter  defect,  which  is  quite  distinct  from  the  other 
aberrations  that  have  to  do  more  with  the  sharpness  of  the  image,  will 
be  explained  more  fully  in  the  following  investigation. 

286.  Image-Points  regarded  as  lying  on  the  Chief  Rays.  If  there 
were  collinear  correspondence  between  Object-Space  and  Image- 
Space,  the  image  of  an  object-plane  a  perpendicular  at  M  to  the 
optical  axis  of  the  centered  system  of  spherical  surfaces  would 
not  only  be  reproduced,  point  by  point,  in  the  image-plane  a'  perpen- 
dicular to  the  optical  axis  at  the  point  M',  which,  by  GAUSS'S  Theory, 
is  conjugate  to  the  axial  object-point  M,  but  the  image  would  be  in 
every  respect  precisely  similar  to  the  object.  The  actual  rays,  how* 
ever,  being  subject,  as  we  say,  to  aberrations,  pursue  routes  which, 
in  general,  are  quite  different  from  the  paths  that  they  would  take  if 
the  image  were  ideal;  so  that,  for  example,  an  outgoing  ray,  proceed- 
ing from  an  object-point  P  in  the  plane  a-,  and  traversing  the  optical 
system,  will  emerge  finally  and  cross  the  image-plane  a'  at  a  point  P', 
which  will  be  identical  with  the  GAUssian  image-point  only  under 
exceptional  circumstances.  Moreover,  another  ray  proceeding  from  the 
same  object-point  P  will  generally  determine  a  different  point  Pr  in 
the  image-plane  a'.  This  latter  difficulty  may  be  partially  overcome 
by  the  use  of  a  very  small  stop,  whereby  the  effective  rays  emanating 
from  the  object-point  P  are  all  comprised  within  the  limits  of  a  very 
narrow  bundle,  all  the  rays  of  which  have  nearly  the  same  inclinations 
and,  consequently,  cross  the  image-plane  at  approximately  the  same 
point,  so  that  we  do  have  there  in  a  certain  sense  a  more  or  less  in- 
distinct image  of  P. 

Taking  the  more  general  case,  and  one  that  is,  in  fact,  very  common 
in  actual  optical  instruments,  let  us  suppose  that  this  stop  is  inter- 
posed somewhere  in  the  interior  of  the  optical  system,  with  its  centre 
on  the  optical  axis  at  a  point  which  we  shall  designate  by  the  letter  0, 
and  which  coincides  with  the  point  where  paraxial  rays,  which  in 
the  Object-Space  go  through  the  centre  M  of  the  Entrance-Pupil  (see 
§  257),  cross  the  optical  axis  in  their  progress  through  the  medium  in 
which  the  stop  is  situated.  As  has  been  already  explained  (§  258), 
the  chief  ray  emanating  from  the  object-point  P  is  that  one  of  the  bun- 
dle which  leaves  P  in  such  a  direction  that  it  will,  in  traversing  the 


§  287.] 


Theory  of  Spherical  Aberrations. 


417 


medium  where  the  stop  is,  go  through  the  centre  0  of  the  stop.  The  path 
of  this  ray  will  lie  in  the  meridian  plane  containing  the  object-point 
P,  which  is  here  the  plane  of  the  diagram  (Fig.  138).  If  the  image- 


M 


M  JET 


M' 


FIG.  138. 

DISTORTION  OF  THE  IMAGE,  xx  represents  the  Optical  Axis  of  a  centered  system  of  spherical 
surfaces.  The  position  of  the  stop-centre  is  marked  by  O.  The  Object-Plane  and  the  Image-Plane 
are  designated  by  <r,  a7.  The  straight  lines  with  arrow-heads  show  the  directions  of  portions  of  the 
path  of  the  chief  ray  which  has  its  origin  at  the  Object-  Point  P  (or  Q)  in  the  Object-Plane  a. 

MP=  i?  =  MQ  -  y,    MQ1  =  y,    M'P*  =  »?',    Q'P1  =  &yr, 


plane  a'  is  supposed  to  be  occupied  by  a  screen,  what  actually  appears 
on  this  screen  may  now  be  called  the  practical  image  of  the  plane 
object  perpendicular  to  the  optical  axis  at  M.  Immediately  around 
the  axial  image-point  M'  there  will  be  other  sharp  image-points,  but 
at  a  little  distance  from  the  axis  we  shall  have  image-spots  instead  of 
image-points,  and  these  images  will  be  more  and  more  indistinct,  the 
farther  they  are  from  the  axis.  If  the  diameter  of  the  stop  is  reduced, 
the  effect  will  be  to  diminish  the  dimensions  of  the  image-spots,  and 
as  a  limiting  case  we  may  even  suppose  that  the  stop  contracts  into 
a  mere  point  or  pinhole-opening  at  0,  so  that  only  the  chief  rays  ema- 
nating from  the  points  of  the  object-plane  a  succeed  in  getting  past  the 
stop.  These  chief  rays,  which  constitute  a  sort  of  skeleton  of  the 
bundles  of  effective  rays,  will  determine  by  their  intersections  with 
the  image-plane  a'  the  positions  in  this  plane  of  the  image-points 
corresponding  to  the  points  of  the  object-plane  o\  Thus,  in  this  view 
of  the  matter,  the  point  P',  where  the  chief  ray  emanating  from  P 
finally  crosses  the  image-plane  <r',  is  to  be  considered  as  the  image  of 
the  object-point  P. 

287.  Measure  of  the  Distortion.  The  position  in  the  image-plane 
a'  of  the  point  Qf,  which,  by  GAUSS'S  Theory,  is  conjugate  to  a  point 
Q  in  the  object-plane  cr,  is  denned  by  the  equation: 

M'Q'  =  Y-MQ, 

where  Y  denotes  the  magnitude  of  the  Lateral  Magnification  of  the 
optical  system  for  the  pair  of  conjugate  axial  points  M,  M'  '.  If  we 

28 


418  Geometrical  Optics,  Chapter  XII.  [  §  288. 

suppose  that  the  point  Q  coincides  with  the  object-point  P,  and  if 
we  put: 

MQ  =  y  =  MP  =  17,     M'Q'  =  y', 

the  formula  above  may  be  written  as  follows: 

y'  =  7-77. 

The  Distortion  of  the  image  of  the  point  P  is  measured  by  the 
aberration 

8y'  =  Q'p>  =  M'P'  -  M'Q'  =  rj'  -  yf. 

An  image  which  is  free  from  distortion,  and  which,  therefore,  is  exactly 
similar  to  the  object  in  its  entire  extent,  is  called  orthoscopic,  or  "angle- 
true",  because  straight  lines  are  reproduced  as  straight  lines  and 
homologous  angles  in  the  object  and  image  are  equal.  A  lens  which 
casts  an  orthoscopic  image  is  called  a  "rectilinear"  lens. 

ART.  92.     CONDITIONS  OF  ORTHOSCOPY. 

288.  General  Case :  When  the  Centres  of  the  Pupils  are  Affected 
with  Aberrations.  In  the  diagram  (Fig.  138)  the  optical  axis  of  the 
system  is  represented,  but  the  actual  refracting  surfaces  are  not  shown. 
The  axial  points  M,  M'  are  the  centres  of  the  Entrance-Pupil  and  Exit- 
Pupil,  respectively,  and  0  designates  the  position  of  the  stop-centre. 
The  chief  object-ray  proceeding  from  the  point  P  of  the  object-plane 
<7  crosses  the  axis  (really  or  virtually)  at  a  point  L,  which,  in  general, 
will  not  be  very  far  from  the  point  M,  since,  during  its  progress  through 
the  system,  this  ray  must  pass  (really)  through  the  stop-centre  0. 
Emerging  after  refraction  at  the  last  surface,  this  same  ray  will  finally 
cross  the  axis  (really  or  virtually)  at  a  point  L',  and  determine  by  its 
intersection  with  the  image-plane  a'  the  point  P',  which  we  have  agreed 
to  consider  as  the  image  of  the  object-point  P.  If 

6  =  ZMLP,     6'  =  /.M'L'P' 

denote  the  slope-angles  of  the  chief  ray  before  and  after  passing  through 
the  optical  system,  we  have  from  the  diagram: 

MP  =  LM-tan  6,     M'P'  =  L'M'-tan  6'. 

If  A,  A'  designate  the  vertices  of  the  first  and  last  spherical  surfaces 
of  the  system,  and  if  we  put 

AM=u,     A'M'=ur,     AM=u,     A'M'  =  u', 


§  288.]  Theory  of  Spherical  Aberrations.  419 

then,  since, 

LM  =  LM  +  MA  +  AM  =  u  -  u  -  6u, 
L'M'  =  L'M'  +  M'A'  +  A'M'  =  u'-u'-  Su', 
we  obtain  : 

rf  _  u'  -  u'  -  du'    tanJT 
rj          u  —  u  —  du       tan  6 

Now  if  the  image  is  to  be  free  from  distortion,  the  point  Pf  must  coin- 
cide with  the  ideal  image-point  Q'',  that  is,  rj'  =  Mf  P'  must  be  iden- 
tical with  yr  =  M'Q',  which  means  that  we  must  have: 


and,  hence,  the  Condition  of  Orthoscopy,  which  requires  that  all  pairs 
of  conjugate  chief  rays  shall  trace  similar  figures  on  the  object-plane 
and  image-plane,  may  be  expressed  by  the  following  formula: 

tan  6'        u  —  u  —  8a 

'          ''y; 


which  involves,  therefore,  not  merely  the  ratio  of  the  tangents  of  the 
slope-angles  6,  0'  of  the  chief  ray,  but  also  the  Longitudinal  Aberrations 
6u,  5u'  at  the  centres  of  the  Pupils. 

If  the  Lateral  Magnification  of  the  system  with  respect  to  the  Pupil- 
Centres  M  ,  M  '  is  denoted  by  Y,  it  may  readily  be  shown,  by  the  aid 
of  formulae  (127)  and  (153),  that  we  have  always  the  following  re- 
lation between  Y  and  Y: 

T7      n     M'M'     i  .     ^ 

=    ''-'> 


where  n,  n'  denote  the  indices  of  refraction  of  the  first  and  last  media. 
Hence,  we  may  also  write  formula  (305)  above  in  the  following  form: 

tan  0'      n     u'  —  u'      u  —  u  —  8u       i 


tan  6   ~  n'     u  —  u     u'  —  u'  —  du'    Y 

In  case  the  object-point  P  is  infinitely  distant,  the  image-plane  cr' 
will  coincide  with  the  secondary  focal  plane  e'  and  the  point  M'  will 
coincide  with  the  secondary  focal  point  £'.  Under  these  circum- 
stances, we  find  : 

n  M'E'  i 


_  _  _ 

tan6  "  n'  '  M'E'  -  M'L'  *  K 

so  that  in  this  instance  the  Longitudinal  Aberration  of  the  ray  at  the 


420  Geometrical  Optics,  Chapter  XII.  [  §  289. 

Entrance-Pupil  does  not  matter.  And  if,  as  in  the  case  of  the  Tele- 
scope, both  object  and  image  are  infinitely  distant,  so  that  Er  is  also 
the  infinitely  distant  point  of  the  optical  axis,  then: 

tan  6'      n      i 

tenT  =  »'  '  Y  "  C°nStant' 

and  the  Condition  of  Orthoscopy  in  this  special  case  is  independent 
of  the  aberrations  at  both  Pupil-Centres. 

It  is  sometimes  stated  that  the  constancy  of  the  tangent-ratio 
tan  0'/tan  6,  known  as  AIRY'S  Tangent-  Condition,1  is  the  necessary 
and  sufficient  condition  of  freedom  from  distortion;  but,  as  M.  VON 
RoHR2  has  pointed  out,  this  is  evidently  by  no  means  the  case  except 
under  special  circumstances.  For  example,  if,  as  is  the  case  with  a 
certain  class  of  Photographic  Objectives,  the  stop  coincides  with  the 
Exit-Pupil,  so  that  the  three  points  designated  by  0,  M',  L'  are  all 
coincident,  then  for  an  infinitely  distant  object-point,  just  as  also  in 
the  case  of  the  Astronomical  Telescope,  the  constancy  of  the  Tangent- 
Ratio  is  the  condition  of  orthoscopy. 

289.  Case  when  the  Pupil-Centres  are  without  Aberration.  If  the 
stop  is  placed,  say,  between  the  kth  and  the  (fc  +  i)th  spherical  sur- 
faces, the  optical  system  will  be  divided  into  two  parts,  an  anterior 
part  (I)  composed  of  the  first  k  spherical  surfaces  and  a  posterior  part 
(II)  composed  of  all  the  spherical  surfaces  after  the  kth.  If  the  part 
(I)  is  spherically  corrected  for  the  centre  of  the  Entrance-Pupil  and 
the  centre  of  the  stop,  and  if,  similarly,  the  posterior  part  is  spherically 
corrected  for  the  centre  of  the  stop  and  the  centre  of  the  Exit-Pupil, 
the  chief  rays  which  are  obliged  to  go  through  O  will  also  go  through 
the  points  M,  M' .  In  this  case  the  Longitudinal  Aberrations  at  M ,  M' 
will  vanish,  that  is,  du  =  du'  =  o;  and  now  the  condition  of  orthos- 
copy is: 

tan  0'      n      i 

~r-  =  —  •  v  =  constant, 
tan  8       n'    Y 

This  is  AIRY'S  Tangent-Condition  above-mentioned,  viz.,  that  the 
ratio  of  the  tangents  of  the  slope-angles  of  every  pair  of  conjugate 

1  G.  B.  AIRY:  On  the  spherical  aberration  of  the  eye-pieces  of  telescopes:  Camb.  Phil. 
Trans.,  Ill  (1830),  1-64.  This  paper  was  published  separately  in  Cambridge  three  years 
before  it  appeared  in  the  Phil.  Trans. 

2M.  VON  ROHR:  Beitrag  zur  Kenntniss  der  geschichtlichen  Entwicklung,  der  An- 
sichten  ueber  die  Verzeichnungsfreiheit  photographischer  Objektive:  Zft.  f.  Instr.,  xviii 
(1898),  4-12.  See  also  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen 
Aberrationen:  Chapter  V  of  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904), 
edited  by  M.  VON  ROHR;  see  page  241. 


§  291.] 


Theory  of  Spherical  Aberrations. 


421 


chief  rays  must  be  the  same.  The  requirement  of  spherical  correction 
of  the  stop-centre  for  the  two  parts  (I)  and  (II)  of  the  optical  system 
is  called  by  VON  ROHR  the  Bow-SuxxoN  Condition.1  When  both  condi- 
tions are  satisfied,  the  points  M,  Mr  are  called  the  Orihoscopic  Points 
of  the  system. 

290.  The  Two  Typical  Kinds  of  Distortion.     If  when  the  two  partial 
systems  are  spherically  corrected  with  respect  to  the  stop-centre  the 
ratio  tan  9'  :  tan  8  is  not  constant,  the  magnification  of  the  image 
close  to  the  optical  axis  will  be  constant,  but  out  towards  the  edges 
there  will  be  distortion.     For  example,  if  with  increasing  values  of  6, 
the  ratio  tan  0'  :  tan  0  also  increases,  the  magnification  rj'/y  will  in- 
crease towards  the  margin  of  the  field,  so  that  spaces  of  equal  area  in 
the  object-plane  will  appear  distorted  in  the  image  into  spaces  of 
gradually  increasing  size  as  we  go  out  from  the  axis.     If  the  object 
consists  of  a  network  of  two  mutually  perpendicular  systems  of  equi- 
distant parallel  lines,  as  in  Fig. 

139  (a)j  the  image  will  appear 
as  in  Fig.  139  (b).  This  case 
is  known  as  "  Cushion- Shaped 
Distortion",  sometimes  called 
also  Positive  Distortion.  On 
the  other  hand,  if  tan  0' :  tan  0 
decreases  as  the  slope-angle 
increases,  the  magnification  t\  \y  will  diminish  out  from  the  centre  of 
the  image;  and  then  we  have  the  case  known  as  "Barrel-Shaped,  Dis- 
tortion", or  Negative  Distortion  (Fig.  139  (c)). 

291.  Distortion  when  the  Pupil-Centres  are  the  Pair  of  Aplanatic 
Points  of  the  System.     If  the  points  M,  M'  are  the  pair  of  Aplanatic 
Points  of  the   system,  they  must  satisfy  the  Sine-Condition,  viz., 
sin0'/sin  0  =  constant;  and  since  this  condition  is  necessarily  opposed 
to  the  Tangent-Condition,  the  image  in  this  case  will  be  distorted  in  such 
fashion  that  r/will  be  less  than  the  ideal  value  yf.     Moreover,  since  the 
tangent  of  an  angle  increases  faster  than  its  sine,  the  difference  yr  —  rj' 
will  increase  as  y  increases,  and  therefore  the  distortion  will  be  "barrel- 
shaped"  (Fig.  139  (c)).     If  the  object  consists  of  two  sheaves  of  hyper- 
bolae resembling  Fig.  139  (6),  and  if  M,  M'  are  the  pair  of  Aplanatic 
Points,  the  image  in  this  case  will  be  the  two  systems  of   parallel 
straight  lines  (Fig.  139  (a)).     This  is  the  test  which  ABBE  invented  to 

1  R.  H.  Bow:  On  Photographic  Distortion:  Brit.  Journ.  of  Photography,  VIII  (1861), 
pages  417-419  and  440-442. 

X.  SUTTON:  Distortion  Produced  by  Lenses:    Phot.  Notes,  VII  (1862),  No.  138,  3-5. 


b 

FIG.  139. 
SHOWING  THE  TYPICAL  KINDS  OF  DISTORTION. 


422  Geometrical  Optics,  Chapter  XII.  [  §  292. 

determine  whether  a  microscope  fulfils  the  Sine-Condition,  and  which 
was  alluded  to  in  §  279. 

ART.  93.     DEVELOPMENT  OF  THE  APPROXIMATE  FORMULA  FOR  THE  DIS- 

TORTION-ABERRATION IN  CASE  THE  SLOPE-ANGLES  OF 

THE  CHIEF  RAYS  ARE  SMALL. 

292.  Here  we  have  to  do  with  the  image  of  an  extensive  object 
formed  by  infinitely  narrow  bundles  of  rays;  that  is,  with  a  large  visual 
field  and  very  small  aperture.  Accordingly,  in  the  series-develop- 
ment of  the  aberrations  of  the  3rd  order  (see  Art.  80,  especially  §  259), 
the  terms  involving  the  co-ordinates  y^  zl  will  be  negligible,  whereas 
the  most  important  term  will  be  the  one  involving  y\.  Since  the  image 
is  determined  entirely  by  the  chief  rays,  and  since  the  path  of  every 
such  ray  lies  in  a  meridian  plane  of  the  optical  system,  the  image-point 
P'm  determined  by  the  intersection  of  the  chief  ray  with  the  image- 
plane  <jm  (§  254)  will  lie  in  the  meridian  plane  which  contains  the  path 
of  the  chief  ray;  consequently,  here  the  s-aberration  (§256)  is  equal 
to  zero,  i.  e.,  dz'm  =  o,  and  we  have,  therefore,  merely  to  develop  the 
expression  for  the  ^-aberration  : 


The  method  of  obtaining  this  development,  which  is  given  below,  is 
the  same  as  that  given  by  KOENIG  and  VON  RoHR.1 

According  to  GAUSS'S  Theory,  the  Lateral  Magnification  Y  of  a 
centered  system  of  m  spherical  surfaces,  with  respect  to  the  pair  of 
axial  conjugate  points  Mlt  M'm,  is  given  by  formula  (93)  of  Chap.  VI, 
as  follows: 


where  3^  =  1^  =  Mf^M^  ym  =  M'mQ'm,  uk  =  A^^,  uk  =  AkM'k. 
Accordingly,  if 

M'mP'm  =  C     Q'mP'm  =  dy'M, 
we  have: 


%  =  n"      TT  **  . 

y'm  ' 


or,  snce 

y\  =  fi  =  C 

1  See  Die  Theorie  der  optischen  Instrumente  (Berlin,  1904),  edited  by  M.  VON  ROHR; 
Chapter  V,  Die  Theorie  der  sphaerischen  Aberrationen,  by  A.  KOENIG  and  M.  VON  ROHR, 
pages  241-246. 


§292.] 
and 


Theory  of  Spherical  Aberrations. 


423 


we  obtain 


We  proceed,  therefore,  to  develop  an  expression  for  the  quotient 


n'ju 


This  expression  relates  to  the  kth  spherical  surface,  but  for  the  present 
it  will  be  convenient  to  drop  the  subscripts.  The  subscripts  are  like- 
wise omitted  from  the  letters  in  the  diagram  (Fig.  140),  which  repre- 


FiG.  140. 

FIGURE  USED  IN  THE  DERIVATION  OF  THE  DISTORTION-ABERRATION  FORMULA.  The  figure 
represents  the  path  of  a  chief  ray  incident  on  the  £th  surface  of  a  centered  system,  of  spherical 
refracting  surfaces. 


sents  the  path  of  the  chief  ray  before  refraction  at  the  kth  surface, 
the  vertex  and  centre  of  which  are  designated  accordingly  by  the 
letters  A  and  C,  respectively.  This  ray  crosses  the  axis  at  the  point 
designated  in  the  figure  by  L  and  is  incident  on  the  kth  surface  at  the 
point  designated  by  B.  The  place  where  the  transversal  plane  of  the 
<7-system  belonging  to  the  medium  immediately  in  front  of  the  kth 
surface  is  cut  by  the  optical  axis  is  marked  by  the  letter  M,  and  the 
point  where  the  ray  crosses  this  plane  is  designated  by  P.  Finally, 
the  foot  of  the  perpendicular  let  fall  on  the  optical  axis  from  the  inci- 
dence-point B  is  designated  by  D.  Let  us  also  use  the  following 
symbols : 

A  C  =  r,     AM  =  u,     AL  =  v,     MP  =  rj, 


424  Geometrical  Optics,  Chapter  XII.  [  §  292. 

The  function  rj  =  M P  changes  its  sign  when  the  slope-angle  0  (or  the 
central  angle  <|>)  changes  sign,  and  hence  this  function  may  be  developed 
in  a  series  of  odd  powers  of  6  (or  c|>).  Since  the  central  angle  <)>  remains 
constant  during  the  refraction,  it  will  be  better  to  take  this  angle  as 
the  independent  variable.  Accordingly,  assuming  that  the  central 
angles  <|>  of  the  chief  rays  are  all  so  small  that  we  can  neglect  the  powers 
of  these  angles  above  the  3rd,  we  may  write: 

77  =  /<)>  +  w<(>3, 

where  the  co-efficients  denoted  by  /,  m  are  constants,  the  magnitudes 
of  which  will  depend  on  the  positions  of  the  transversal  planes  o-,  <r 
belonging  to  the  medium  which  the  ray  is  here  traversing. 
From  the  figure,  we  see  that: 

MP      LM     LA  +AM 
DB~  LD~  LA+AD' 

and,  accordingly,  if  we  remark  that: 


.2* 


DB  =  r-  sin  <)>,     AD  =  2r-  sin2  z  , 


we  obtain: 

(v  — 


v  —  2r-sin  - 


,       . 
(307) 


In  this  formula  we  may  substitute  for  v  the  development: 

v  =  u  +  c<|>2, 

where  u  —  AM  denotes  the  abscissa  of  the  point  M  where  paraxial 
rays,  which  in  the  first  medium  go  through  the  centre  of  the  Entrance- 
Pupil,  cross  the  axis  before  refraction  at  the  surface  here  under  con- 
sideration; and  where  c  denotes  an  undetermined  coefficient.  Ob- 
viously, 


is  the  expression  for  the  Longitudinal  Aberration  of  the  chief  ray 
before  refraction  at  this  spherical  surface  (see  §  263).  Making  the 
substitution  above  mentioned,  and  equating  the  two  expressions  of  the 
function  77,  we  obtain  after  clearing  fractions,  expanding  the  trigono- 
metric functions  in  series,  and  arranging  according  to  powers  of  <j>, 
the  following  equation  : 


§  292.]  Theory  of  Spherical  Aberrations.  425 

and  since  this  equation  must  be  true  for  all  the  chief  rays,  that  is, 
for  all  values  of  the  central  angle  <|>  (as  far  as  the  extreme  value  per- 
mitted by  this  approximation)  ,  we  may  equate  to  zero  the  co-efficients 
of  <|>  and  <|>3;  whereby  the  magnitudes  of  the  co-efficients  /,  m  are 
determined  as  follows: 

.      u  —  u 

I  =  —  -  r 


U   —  U 


—  U    3  /_!_     ,  U  C  I     1 

u         \2ru  +  u(u-ur2~~6r2J 


Substituting  these  expressions  for  the  co-efficients  J,  m  in  the  series- 
development  of  the  function  77,  and  at  the  same  time  using  the  in- 
variant-relation obtained  by  combining  formulae  (270)  : 

n(u  -  u)      ri(u'  -  u') 

J  —  J  =  -  -  •  =  ---  f—.  -  •  , 

uu  uu 

we  derive  the  following  formula  : 


Similarly,  for  the  ray  refracted  at  this  surface,  we  obtain  the  corre- 
sponding formula  for  n'rj'/u'  by  merely  priming  the  letters  u  and  c  on 
the  right-hand  side  of  equation  (308).  Doing  this,  and  dividing  the 
latter  equation  by  the  former,  we  obtain  : 

n'vU  jjo  f    !     A    l  T  A  nC  \ 

-7  =  1+  rd>  1  —  A-  —  -o,  _      yr  A—  2  f? 
nrju'  I  ^r    u      r(J  -  J)     u2  J 

Now 


-  o~       --       '**  —  » 
u  u  2  nu 

as  we  found  in  §  263.     Moreover,  according  to  formula  (77), 


u  n 

Thus,  we  obtain: 

AI       J2  A' 

A  ---  ^  --  ,  -A  — 
W        J  —  J     UU 

It  h  =  DB  denotes  the  incidence-height  of  the  chief  ray  corre- 
sponding to  the  central  angle  <|>,  we  may,  neglecting  magnitudes  above 
the  3rd  order,  put 

h2  =  r. 


426  Geometrical  Optics,  Chapter  XII.  [  §  292. 

Re-introducing  the  subscripts,  we  shall  write,  therefore,  the  above 
equation  as  follows: 

***"*.   -i-^f^A^V    -£-Jl-\    \ 

»L_iliUX  2  \  rk     \n)k     Jk  -  Jk     \nujk\ 

Before  substituting  this  expression  in  the  equation  for  dy'm,  we  shall 
put  it  in  a  form  rather  more  convenient  for  actual  use.  Thus,  if  /^ 
denotes  the  incidence-height  at  the  first  spherical  surface  of  a  paraxial 
object-ray  proceeding  from  the  axial  object-point  Mv  and  if  hk  denotes 
the  incidence-height  at  the  &th  surface  of  this  same  paraxial  ray,  we 
may,  without  neglecting  magnitudes  of  the  3rd  order,  use  the  relation 
given  by  formula  (155)  of  Chapter  VIII,  viz.: 


in  which  case  the  expression  on  the  right-hand  side  of  the  above  equa 
tion  may  be  transformed  as  follows: 


Substituting  this  expression  in  the  formula  for  the  aberration  8y'm,  we 
obtain  : 


Thus,  the  condition  that  8y'm  =  o,  or  that  there  shall  be  no  distortion 
of  the  image-point  P'm  is  as  follows  : 


Formula  (309)  may  be  put  also  in  a  different  form.     Thus,  since 
(as  may  be  easily  shown,  see  §  126)  we  have  the  following  relation: 

_L     J  ~  J   I    1     i    J_ 
nu  ~       J      '  r     n      J    nu' 

we  may  substitute  this  expression  within  the  large  brackets,  whereby, 
after  simple  reductions,  we  obtain: 


§  293.]  Theory  of  Spherical  Aberrations.  427 

Finally,  according  to  the  Law  of  ROBERT  SMITH  (§  194),  the  rela- 
tion between  the  conjugate  ordinates  ylt  ym  may  be  expressed  evi- 
dently as  follows: 

nmymhm 


and,  moreover,  we  have  also: 

*,«- 

and 


Jl  -  Jl      n^u,  -  «!>' 
And,  hence  the  formula  above  may  be  written: 


whence  it  is  seen  how  the  Distortion-Aberration  6^  is  proportional 
to  the  cube  of  the  ordinate  yv 

ART.  94.     THE  DISTORTION-ABERRATION  IN  SPECIAL  CASES. 

293.     Case  of  Single  Spherical  Refracting  Surface. 
When  the  optical  system  is  composed  of  a  single  spherical  surface, 
formula  (309)  gives  for  the  Distortion-Aberration 


where,  for  the  sake  of  brevity,  we  write: 


r         n  nu 

or 

2 

n'          _  nf  +  n      zn      n'       n 
n  —  n'  u2      ""  ru  ""  ru      r2  ' 

If  the  image  is  to  be  free  from  distortion,  we  must  have  by'  =  o;  which 
implies  here  one  of  two  things:  Either  J  =  o,  or  else,  T  =  o.  If 
J  =  o,  then  u  =  u'  =  r;  which  means  in  this  case  that  the  stop-centre 
coincides  with  its  image  at  the  centre  C  of  the  spherical  surface,  and 
under  these  circumstances  the  image  will  be  free  from  distortion  for 
all  object-distances. 


428  Geometrical  Optics,  Chapter  XII.  [  §  294. 

If  the  stop-centre  is  not  at  C,  the  condition  that  the  image  shall 
be  without  distortion  is  T  =  o,  or 


whence  it  is  seen  that  for  a  given  position  of  the  centre  L  of  the  stop, 
that  is,  for  a  given  (real)  value  of  the  abscissa  u,  there  will  always  be 
one  certain  object-distance  u  for  which  the  spherical  surface  will  give 
an  image  free  from  distortion. 

If,  on  the  other  hand,  the  object-distance  u  is  given,  we  shall  find 
that  there  are  always  two  positions  of  the  stop-centre  that  will  give 
an  image  without  distortion;  that  is,  corresponding  to  a  given  value 
of  u,  we  obtain  two  values  of  u,  viz.: 


n         \n'(n 
r      V~ 


I  i       I  n         \n'(n  + »')       nn 


u      n  +  n'  |  r     '  \        rw  r1 

If  these  two  values  of  «  are  to  be  real,  u  and  r  must  have  the  same 
signs  and 

r          n 

u      n  -f-  n' ' 

If  the  object-point  M  coincides  with  the  aplanatic  point  Z  (§  207), 
then  u  —  u  =  (n  -f-  n')r/n,  which  means  that  the  stop-centre  also  co- 
incides with  the  aplanatic  object-point  Z;  this,  however,  would  have 
no  practical  meaning,  and,  hence,  the  distortion  cannot  be  abolished 
when  the  object-point  M  coincides  with  the  aplanatic  point  Z.  This 
is  in  agreement  with  the  results  which  we  found  when  we  were  invest- 
igating the  Sine-Condition  (§276;  cf.  §289). 

294.     Case  of  Infinitely  Thin  Lens. 

The  distortion  produced  by  an  optical  system  consisting  of  an 
Infinitely  Thin  Lens  may  be  investigated  by  a  method  precisely  simi- 
lar to  that  used  in  the  case  of  the  Longitudinal  Aberration  (see  §  268). 
If  here  also  we  use  the  same  special  Lens-Notation  as  was  employed 
there,  the  expression  within  the  large  brackets  in  formula  (309)  may 
be  put  equal  to  —  <pX,  where  <p  denotes  here  the  reciprocal  of  the 
Primary  Focal  Length  (/)  of  the  Lens,  and  where  the  function  denoted 
by  X  will  have  the  following  expression: 


n  —  i  n 

-2 


n  +  i  \ 

n       ) 


I—    ~Ti?        -  -      -  — 

(n  —  i)  n  n  —  i  n  —  i 


§  295.]  Theory  of  Spherical  Aberrations.  429 

The  condition  that  X  shall  be  a  minimum  for  given  values  of  <p,  x 
and  ac  will  be  found  to  be: 

_  3(11  +  i)  n  +  i  n(2n  +  i) 

=  *  *-  *  i~  >- 


V.     ASTIGMATISM  AND  CURVATURE  OF  THE  IMAGE. 
ART.  95.     THE  PRIMARY   AND    SECONDARY   IMAGE-SURFACES. 

295.  In  the  imagery  of  extended  objects  by  means  of  narrow 
bundles  of  rays  whose  chief  rays  all  meet  at  a  prescribed  point  on  the 
optical  axis  of  the  centered  system  of  spherical  surfaces,  there  will, 
in  general,  be  astigmatic  deformation  of  the  bundles  of  image-rays; 
in  consequence  whereof  to  an  object-point  P  lying  outside  the  axis 
there  will  correspond,  not  a  sharp  image-point,  but  two  short  image- 
lines  perpendicular  to  the  chief  ray  of  the  bundle  at  the  so-called  I. 
and  II.  Image-Points  S'  and  3'  (see  Chapter  XI).  Thus,  in  case  the 
image-rays  are  received  on  a  focussing-screen,  the  image  of  the  object- 
point  as  'seen  on  the  screen  will  generally  be  a  small  patch  of  light 
corresponding  to  the  cross-section  of  the  bundle  of  image-rays  at  that 
place,  the  dimensions  of  which,  in  one  direction  at  least,  will  always 
be  comparable  with  the  diameter  of  the  narrow  stop;  so  that  such  an 
image  formed  by  an  astigmatic  bundle  will  always  be  more  or  less 
blurred  and  indistinct,  and  not  to  be  compared  in  this  respect  with 
the  sharp  image  which  is  obtained  when  the  object-point  is  on  the 
axis.  The  farther  the  object-point  is  from  the  axis,  the  more  pro- 
nounced this  defect  will  be.  In  two  special  positions  of  the  focussing- 
screen  the  image  will  be  deformed  into  a  short  line,  which  is  vertical, 
say,  for  one  of  the  positions,  and  horizontal  for  the  other  position 
—  corresponding  to  the  places  of  the  two  image-lines  of  the  astigmatic 
bundle  (§230).  Somewhere  between  these  two  positions  the  bundle 
of  rays  will  have  its  narrowest  cross-section,  which,  in  the  case  of  a 
centered  system  of  spherical  surfaces,  will  be  approximately  circular 
in  form.  This  is  the  place  of  the  so-called  "Circle  of  Least  Confu- 
sion" (§  244)  —  a  somewhat  misleading  phrase,  inasmuch  as  the  con- 
vergence of  the  rays  in  either  of  the  two  image-lines  is  of  a  higher 
-order.  However,  we  do  obtain  here  perhaps  the  nearest  approach  to 
a  true  image  of  the  object-point. 

If  on  every  chief  image-ray  corresponding  to  such  points  of  the 
object  as  are  contained  in  a  meridian  plane  of  the  optical  system,  we 
mark  the  I.  and  II.  image-points  S'  and  3',  the  loci  of  these  two  sets 
of  image-points  will  be  two  curved  lines  which  touch  each  other  at 


430  Geometrical  Optics,  Chapter  XII.  [  §  296. 

their  common  vertex  M'  where  they  both  cross  the  optical  axis;  so 
that  this  point  M'  is  an  accurate  image  of  the  axial  object-point  M. 
Assuming  that  the  points  of  the  object  lie  on  a  surface  of  revolution 
described  around  the  optical  axis,  for  example,  in  a  plane  perpen- 
dicular to  the  optical  axis,  we  readily  perceive  that  these  curves  are 
the  traces  in  the  meridian  plane  of  the  /.  and  II.  I  mage- Surf  aces, 
which  latter  will,  therefore,  be  generated  by  revolving  these  curves 
around  the  optical  axis  as  axis  of  rotation.  One  of  these  surfaces  will 
contain  all  the  I.  Image-Lines,  and  the  other  will  contain  all  the  II. 
Image-Lines.  A  third  surface  of  revolution,  lying  between  these  two, 
will  contain  the  circles  of  least  confusion. 

Even  if  the  bundles  of  image-rays  were  made  stigmatic,  so  that  the 
I.  and  II.  image-surfaces  coincided  into  a  single  image-surface  corre- 
sponding with  the  object-surface  point  by  point,  the  image  would,  in 
general,  still  be  curved,  so  that  if  the  image-rays  were  received  on  a 
flat  screen  perpendicular  to  the  optical  axis,  the  definition  of  the  image- 
points  as  seen  on  the  screen  would  still  be  more  or  less  faulty  depending 
on  the  degree  of  curvature  of  the  image-surface.  In  the  case  of  most 
optical  instruments,  and  especially  in  the  case  of  the  photographic 
objective  and  of  the  lantern-projection  system,  a  flat  image  is  an 
essential  requirement;  so  that  closely  connected  with  the  abolition  of 
the  astigmatism  of  the  oblique  bundles  of  image-rays  is  the  removal  of 
the  so-called  "aberration  of  curvature"  of  the  image-surface. 

The  methods  employed  in  the  following  investigation  will  be  found 
to  be  similar  to  the  treatment  of  this  subject  in  KOENIG  and  VON 
ROHR'S  Die  Theorie  der  sphaerischen  Aberrationen.1 

ART.  96.     THE  ABERRATION-LINES,  IN  A  PLANE  PERPENDICULAR  TO  THE 
AXIS,    OF   THE  MERIDIAN   AND    SAGITTAL   RAYS. 

296.  At  the  image-point  M'  (Figs.  141  and  142),  corresponding  to 
the  axial  object-point  M1  let  us  suppose  that  the  GAUSsian  image- 
plane  a'  is  erected.  The  chief  ray  of  the  bundle  of  rays  proceeding 
from  an  object-point  P,  not  on  the  optical  axis,  will,  in  traversing 
the  medium  where  the  infinitely  narrow  stop  is  placed,  go  through  the 
centre  0  of  this  stop,  and,  finally,  after  refraction  at  the  last  surface 
of  the  system,  will  cross  the  axis  (really  or  virtually)  at  a  point  L' 
and  meet  the  image-plane  a'  in  the  point  Pr .  Let  Hf  (Fig.  141)  and 
G'  (Fig.  142)  designate  the  points  where  the  outermost  meridian  ray 

1  Die  Theorie  der  optischen  Instrumente:  Bearbeitet  von  wissenschaftlichen  Mitar- 
beitern  an  der  optischen  Werkstaette  von  CARL  ZEISS;  Bd.  I  (Berlin,  1904),  heraus- 
gegeben  von  M.  VON  ROHR.  V.  Kapitel.  250-265. 


296.] 


Theory  of  Spherical  Aberrations. 


431 


and  the  outermost  sagittal  ray,  respectively,  of  the  infinitely  narrow 
astigmatic  bundle  of   image-rays  cross  the  transversal  plane  at  L' 


AC" 


FIG.  141. 

NARROW  PENCIL  OF  MERIDIAN  IMAGE-RAYS.  The  chief  ray  IJ P'  of  the  pencil  crosses  the  optical 
axis  at  the  point  U ,  and  meets  the  GAUSSian  Image-Plane  <r'  at  P1 '.  H'  U'  is  the  extreme  ray  of  the 
pencil  crossing  at  ff  and  If  the  planes  perpendicular  to  the  optical  axis  at  Lf  and  M1 ',  respect 
ively.  These  rays  intersect  each  other  in  the  I.  Image-Point  Sf,  and  the  locus  of  the  I.  Image- 
Points  S'  is  the  Primary  Image-Curve  whose  centre  of  curvature  with  respect  to  the  point  Aff  is  at 
the  point  A*.  The  chief  ray  meets  a  plane  a"  parallel  to  v'  at  the  point  P".  P'  U'  is  the  aberration 
line  of  the  pencil  of  meridian  rays  in  the  GAUSSiau  Image-Plane  <r'. 

UM*  =uf  —  v',    M'K'  =  &.    M'M"  =  e.    M' P'  =  if.      LM'JJP'  =  9'      Z  L'S'H1  =  rfA'. 


which  is  parallel  to  the  image-plane  cr'.  The  extreme  ray  of  the  pencil 
of  meridian  rays  will  intersect  the  chief  ray  at  the  I.  image-point  5', 
and  will  meet  the  image-plane  <r'  in  a  point  U'  (Fig.  141)  lying  in  the 


FIG.  142. 

NARROW  PENCIL  OF  SAGITTAL  IMAGE-  RAYS.  The  chief  ray  U  P'  of  the  pencil  crosses  the  optical 
axis  at  the  point  if  and  meets  the  GAUSSian  Image-Plane  o"'  at  the  point  P'  .  G'  V'  is  the  extreme 
ray  of  the  pencil  crossing  at  G'  and  V*  the  planes  perpendicular  to  the  optical  axis  at  If  and 
M',  respectively.  _These  rays  intersect  each  other  in  the  II.  Image-  Point  5',  and  the  locus  of  the 
II.  Image-Points  &  is  the  Secondary  Image-Curve,  whose  centre  of  curvature  with  respect  to  the 
axial  Image-Point  M'  is  at  the  point  marked  K'  .  The  chief  ray  meets  a  plane  <r"  parallel  to  o'  at 
the  point/"',  and  the  ray  G'V  meets  this  plane  in  the  point  V"  .  P*  V  is  the  aberration-line  of 
the  pencil  of  sagittal  rays  in  the  GAUSSian  Image-Plane  a'  . 


meridian  plane;  and,  similarly,  the  extreme  ray  of  the  sagittal  pencil 
will  intersect  the  chief  ray  at  the  II.  image-point  3'  and  will  meet 
the  plane  a'  in  a  point  V  (Fig.  142)  in  the  plane  of  the  sagittal  rays. 


432  Geometrical  Optics,  Chapter  XII.  [  §  297. 

The  line-segments   P'U'   and   P'V'   are   the   aberration-lines,  in   the 
image-plane  a',  of  the  meridian  and  sagittal  rays,  respectively;  and  we 
proceed  now  to  obtain  expressions  for  their  magnitudes. 
Evidently,  we  have  the  following  proportions: 

P'U'  _  S'P'      P'V      5'P' 

whence  we  find : 

P'V  =  *I'L'Hi ,    P'V 


where  0'  =  Z.M'L'P'  denotes  the  slope-angle  of  the  chief  image-ray 
of  this  bundle,  and  Y  (Fig.  141)  and  Z  (Fig.  142)  designate  the  feet 
of  the  perpendiculars  let  fall  from  Sr  and  S'  ',  respectively,  on  the  plane 
a'  .  If  here  we  introduce  the  aperture-angles  of  the  meridian  and 
sagittal  pencils,  viz., 

d\r  =  Z  L'S'H',    d\'  =  Z  L'S'G', 

these  angles  being  supposed  here  so  small  that  we  may  neglect  any  terms 
involving  their  squares,  then  : 


L'H'  =         7<ft'»    L'G' 
and,  hence: 

T-:sS?#  *v-i&*'          (3I3) 

297.     Case  when  the  Slope-Angles  of  the  Chief  Rays  are  Small. 

If  R'  =  M'K1  (Fig.  141)  and  R'  =  M'K'  (Fig.  142)  denote  the 
radii  of  curvature  at  the  common  vertex  M'  of  the  I.  and  II.  image- 
surfaces,  and  if  we  neglect  powers  of  the  slope-angles  6'  above  the 
second,  we  may  write: 

M'Y2      _  M'Z2 


Let  Q'  designate  the  position  in  the  plane  o-'  of  the  point,  which,  by 
GAUSS'S  Theory,  is  conjugate  to  the  object-point  P;  then  the  ordinate 
M'Qf  =>  y'  is  of  the  same  order  of  magnitude  as  M'  Pf  and  6;,  and  thus 
it  is  obvious  that  if  we  neglect  powers  of  6'  above  the  second,  we  may 
write  : 

M'Y2  =  M'Z2  =  /; 


§  298.]  Theory  of  Spherical  Aberrations.  433 

and,  hence,  to  the  required  degree  of  exactness,  we  obtain: 

<s'v  —     y     ~^'7  -     y 

2Rf  '  2R'  ' 

Accordingly,  we  derive  the  following  approximate  expressions  for  the 
magnitudes  of  the  aberration-lines,  in  the  GAussian  image-plane  cr', 
of  the  meridian  and  sagittal  rays: 

P'U'=-^,d\>,    P>V'=-^dl>.  (3I4) 

298.  Moreover,  let  a"  be  any  plane  parallel  to  the  GAussian  image- 
plane  o-',  and  at  a  distance  from  it  M'M"  —  e  (say),  and  let  P",  U" 
and  V"  designate  the  points  where  the  rays  Z/S'S',  H'S'  and  G'3', 
respectively,  cross  the  plane  <r";  so  that  P"U"  and  P"V"  will  be  the 
linear  aberrations  in  this  transversal  plane  of  the  meridian  and  sagit- 
tal rays  of  the  astigmatic  bundle  of  image-rays.  Evidently,  if  we 
neglect  the  second  powers  of  the  aperture-angles  d\',  d\'  and  the 
powers  of  the  slope-angle  6'  above  the  second,  we  shall  have: 


If,  therefore,  supposing  that  we  have  d\r  =  d\',  we  wish  to  determine 
the  position  of  the  focussing-plane  a"  somewhere  between  the  I.  and 
II.  image-points  Sf  and  S'  for  which  the  linear  aberrations  P"U"  and, 
P"V"  are  of  equal  magnitudes  but  of  opposite  signs,  the  two  equations 
(3J5)  giye  the  following  formula  for  this  particular  value  of  the 
abscissa  e: 


and,  under  these  circumstances,  we  obtain: 

pnrjn  _  ynpn  _  /!<?*'  (  1_     .  1\ 

4     \R'     R')' 

If  the  bundle  of  rays  is  received  on  a  plane  screen  coinciding  with 
this  position  of  the  plane  an  ',  we  shall  obtain  on  the  screen,  as  was 
stated  above  (§  295),  perhaps  the  nearest  approach  to  a  true  image 
of  the  object-point. 

29 


434  Geometrical  Optics,  Chapter  XII.  [  §  299. 

In  case  the  astigmatism  was  entirely  abolished,  so  that 

R'  =  R', 

by  placing  the  plane  screen  in  the  position  for  which  e  =  yf2  J2R', 
we  should  obtain  on  it  an  actual  point-image  of  the  object-point  P. 
But  it  will  be  remarked  that  the  value  of  e  depends  on  that  of  y\  and 
in  order  to  obtain  point-images  of  the  different  points  of  the  object, 
we  should  have  to  •' 'focus"  the  screen  so  that  its  intersection  with  the 
curved  stigmatic  image-surface  would  contain  the  point  to  be  observed. 

ART.  97.  DEVELOPMENT  OF  THE  FORMULAE  FOR  THE  CURVATURES 

I/R',  !/£'. 

299.  The  Invariants  of  Astigmatic  Refraction.  The  curvatures  at  Mr 
of  the  two  image-surfaces  have  now  to  be  expressed  in  terms  of  the 
curvature  of  the  object-surface  at  M  and  of  the  given  constants  of  the 
centered  system  of  spherical  surfaces.  In  the  development  of  these 
expressions  we  shall  use  ABBE'S  Invariant-Method,  as  given  by  KOENIG 
and  VON  ROHR  in  their  treatise  on  Die  Theorie  der  sphaerischen  Aber- 
rationen. 

In  Chapter  XI,  §§  236  and  240,  we  derived  two  formulae  (246) 
and  (250),  which  may  be  written  as  follows: 


/  cos  a      cos2  a  \         ,  /  cos  a'      cos2  a'  \ 

Q  =  n\  - I  =  n'  I ; —  I , 

V     r  s     )  V     r  s'     /' 

-         /'cos  a      i\         ,/cosa'      I  \ 

Q  =  n  ( —  .  I  =  ri\          -  —  -.  I ; 

V     r          s)  \     r          r/f 


(3i6) 


where  the  functions  denoted  here  by  Q  and  Q,  which  have  the  same 
values  before  and  after  refraction  at  a  given  spherical  surface,  are 
called  the  Invariant-  Functions  of  the  Chief  Ray  of  the  Infinitely  Narrow 
Bundle  of  Rays.  Each  of  these  functions  may  evidently  be  developed 
in  a  series  of  ascending  powers  of  the  central  angle  <|>  of  the  following 
forms  : 


(317) 


wherein  the  coefficients  B,  B,  etc.   are    as    yet    undetermined,  and 
where,  as  usual,  the  terms  involving   powers  of   <|>   higher  than  the 


§  300.]  Theory  of  Spherical  Aberrations.  435 

second  are  neglected.    The  relations,  which  we  wish  to  find,  will  then 
be  given  by  writing: 

B'  -  B  =  o,     B'  -  B  =  o. 

The  easiest  method  of  obtaining  the  expansions  of  Q  and  Q  will  be 
to  develop  the  functions  I  /s,  i  /s  and  cos  a  each  in  a  series  of  ascending 
powers  of  <(>,  and  to  introduce  these  expressions  in  the  formulae  (316) 
above. 

300.    Developments  of  i  fs,  i  /s  and  cos  a  in  a  series  of  powers  of  <|>. 

In  the  diagram  (Fig.  143)  the  straight  line  SB  represents  the  path 


FIG.  143. 

PATH  OF  CHIEF  RAY  OF  PENCIL  OF  MERIDIAN  RAVS  INCIDENT  ON  £TH  SURFACE  OF  CENTERED 
SYSTEM  OF  SPHERICAL  REFRACTING  SURFACES. 

AC=r,    MK=R,    RS=s,   AM=u,    Al,  =  v,    £L=  I,     £JiCA=*$,     £SKM=tyt     £AL,B=9. 

of  the  chief  ray  before  its  refraction  at  (say)  the  kth  spherical  surface. 
In  its  progress  through  this  medium  the  ray  crosses  the  axis  at  the 
point  designated  by  L  and  is  incident  on  the  spherical  surface  at  the 
point  B.  The  point  designated  by  S  is  the  I.  Image-Point  of  the 
astigmatic  bundle  of  rays  in  the  medium  between  the  (k  —  i)th  and 
kth  spherical  surfaces,  and  the  curved  line  MS  represents  the  section 
in  the  meridian  plane  (or  plane  of  the  figure)  of  the  I.  image-surface 
which  is  the  locus  of  the  I.  image-points  S.  The  primes  and  subscripts 
which  naturally  belong  to  these  letters  are  suppressed  for  the  present  ; 
they  will  re-appear,  as  usual,  at  the  end  of  the  investigation.  For 
the  purpose  of  these  developments,  we  shall  employ,  therefore,  the 
following  symbols: 

A  C  =  r,     MK  =  R,     AM  =  u,     AL  =  v,     BS  =  s, 


The  letter  K  is  used  here  to  designate  the  centre  of  curvature  at  M 
of  the  meridian  section  of  the  I.  image-surface. 


436  Geometrical  Optics,  Chapter  XII.  [  §  300. 

From  the  figure  we  obtain  easily  the  following  relation: 
I  __  cos  6  _ 

S      ~  o\£  0<t>' 

u  +  2.R-sin2  —  -  2r  •  sin2 

2  2 

which,  provided  we  neglect  the  powers  of  the  angles  6,  <(>  and  ^  above 
the  second,  may  be  written: 

_e2 

I  "  2 


~s  =         ^        <t>2 

u  +R  —  —  r  — 

2          2 

Moreover,  when  the  angles  <|>  and  \f/  are  infinitely  small,  we  have: 

R$  ML  v  —  u      u  —  u 

^  =    U++  -^i  -   U++  —  -       -—  , 

This  relation,  which  is  strictly  true  in  case  <j>  =  \j/  =  o,  is  also  true 
provided  we  may  put  sin  <|>  =  <(>  and  sin  \j/  =  $,  that  is,  provided  we 
neglect  the  powers  of  these  angles  above  the  first;  and  even  when  we 
retain,  as  here,  the  second  powers  of  these  angles,  we  may  write  : 


(u-u)* 

In  the  same  way,  also: 

Hence,  eliminating  9  and  \l/  from  these  equations,  we  obtain : 

£  r» 


or,  finally: 


The  development  of  the  reciprocal  of  s  =  US  will  obviously  have 
precisely  the  same  form  as  that  obtained  here  for  1/5;  the  only  dif- 
ference being  that  we  shall  have  R  in  place  of  R  in  formula  (318). 

Again,  since 


cf 
cos  a  =  I  — — 


§  302.]  Theory  of  Spherical  Aberrations.  437 

and  since 


we  obtain  : 

-A2<>2  r2./2 


.  (3I9) 

301.    The  expressions  for  the  co-efficients  B,  B  and  B',  B'. 

If  now  we  substitute  in  the  formulae  (316)  the  series-developments 
for  i/s,  i/s  and  cos  a,  as  found  above  in  formulae  (318)  and  (319), 
we  obtain  the  'following  expressions  for  the  co-efficients  B  and  B  in 
formulae  (317): 


7? 

= 


u        uu         nu 

nr     nr* 


These  expressions  can  be  obtained  in  a  more  convenient  form.     Thus, 
by  simple  transformations: 

_r_J!_>«      nS    ±  r  n      n_SSi_J\* 

'  v  ^ 


n        t          u      u 


n 
and  hence: 


(320) 


The  expressions  for  the  co-efficients  B',  B'  will  evidently  have  the 
same  forms  as  the  expressions  found  above  for  B,  B,  and  can  be  ob- 
tained directly  from  formulae  (320)  by  merely  priming  the  symbols 
w,  R,  R  and  u. 

302.  Imposing  now  the  conditions  B'  —  B  =  o  and  B'  —  B  =  o, 
and  at  the  same  time  introducing  the  subscripts  and  employing  ABBE'S 
difference-notation,  we  derive  the  following  formulae  for  the  relations 
between  the  curvatures  of  the  image-surfaces  before  and  after  refract- 


438  Geometrical  Optics,  Chapter  XII  .  [  §  302, 

ion  at  the  kth  surface  of  the  centered  system  of   spherical  surfaces: 


In  the  case  of  a  centered  system  of  m  spherical  refracting  surfaces, 
we  obtain,  therefore,  by  the  usual  method  of  addition  the  following 
convenient  forms  of  the  relations  between  the  curvatures  of  the  object- 
and  image-surfaces: 


Jc=m  1c=m 


(322) 


If  the  bundles  of  object-rays  are  homocentric,  as  will  usually  be  the 
case,  the  radii  Rj_  and  R^  will  be  identical.  For  a  stigmatic  plane 
object  perpendicular  to  the  optical  axis,  we  shall  have  Rl  =  3^  =  oo ; 
in  which  case  the  second  term  on  the  left-hand  side  of  each  of  the 
above  equations  (322)  will  vanish;  and,  if,  moreover,  the  image  is 
formed  in  air  (nm  =  i),  the  expressions  on  the  right-hand  side  of  the 
two  formulae  (322)  will  give  at  once  the  curvature  of  the  two  image- 
surfaces.1 

If,  assuming  the  usual  case  of  a  stigmatic  object,  we  subtract  the 

1  These  formulae,  practically  in  the  same  form  as  they  are  here  given,  were  published 
by  H.  ZINKEN  gen.  SOMMER  in  a  treatise  entitled  Untersuchungen  tieber  die  Dioptrik  der 
Linsen-Systeme  (Braunschweig,  1870);  see  also  an  article  by  the  same  writer  on  the  same 
subject  in  POGG.  Ann.,  cxxii.  (1864),  563-574.  Also,  L.  SEIDEL:  Zur  Dioptrik.  Ueber 
die  Entwicklung  der  Glieder  3ter  Ordnung,  welche  den  Weg  eines  ausserhalb  der  Ebene 
der  Axe  gelegenen  Lichtstrahles  durch  ein  System  brechender  Medien,  bestimmen:  Astr. 
Nachr.,  xliii.  (1856),  Nos.  1027,  1028  and  1029,  paragraph  8. 

H.  CODDINGTON  in  his  celebrated  treatise  on  the  Reflexion  and  Refraction  of  Light 
(Cambridge,  1829)  had  derived  equivalent  formulae  for  the  curvatures  of  both  the  I.  and 
II.  I  mage- Surf  aces  under  the  same  restrictions  as  we  have  here  imposed.  CODDINGTON'S 
methods,  which  are  always  highly  ingenious,  are  employed  in  H.  DENNIS  TAYLOR'S  A 
System  of  Applied  Optics  (London,  1906).  Prior  to  CODDINGTON,  G.  B.  AIRY  had  published 
a  small  volume,  On  the  Spherical  Aberration  of  the  Eye- Pieces  of  Telescopes  (Cambridge, 
1827),  afterwards  reprinted  in  the  Cambridge  Phil.  Trans.,  iii.  (1830),  in  which  he  invest- 
igated the  curvature  of  the  image-surface.  We  must  not  omit  to  refer  also  to  the  invest- 
igations of  P.  BRETON  DE  CHAMP,  published  in  the  Comptes  Rendus  in  1855,  '6  (Tome 
xl..  No.  4,  189-192;  tome  xlii.,  No.  12,  542-545  and  No.  16,  741-744  and  No.  20,  960-963). 

As  to  the  celebrated  formula  published,  in  1843,  by  J.  PETZVAL,  reference  will  be  made 
to  that  in  the  text. 


§  303.]  Theory  of  Spherical  Aberrations.  439 

two  equations  (322),  we  obtain: 

k=m  I*  /    T    \ 

I  I  ,          X~>  ~    k  A         I  *  \  /  \ 

<  -  s:  -  -  2ra™  S  C/FJ?  A  feX'       (323) 

and,  hence,  /&e  condition  of  the  abolition  of  the  astigmatism  of  the  bundles 
vf  image-rays,  viz.,  R'm  =  R'm,  becomes: 


and,  exactly,  as  in  §292,  we  may  employ  here  also  formula  (155)  of 
Chap.  VIII,  viz.: 

;*  A  ai'-^i)=Mi*(^  -•/*), 

whereby  formula  (324)  may  evidently  be  put  in  the  following  form: 


a  formula  of  great  simplicity  and  convenience,  since,  exactly  as  in  the 
case  of  the  formula  for  the  Longitudinal  Aberration  along  the  axis,  it 
enables  us  to  see  distinctly  the  effect  of  each  single  refraction,  and 
thereby  to  ascertain  the  factors  which  have  the  most  influence  on  the 
astigmatism. 

303.  Curvature  of  the  Stigmatic  Image.  If  the  astigmatism  is 
abolished,  we  obtain  for  the  curvature  of  the  image  : 

(326) 

whence  it  is  seen  that  the  curvature  of  the  stigmatic  image  is  independent 
vf  the  position  of  the  stop. 

This  is  the  so-called  "PETZVAL  Formula",  which  was  published,  un- 
fortunately without  proof,  by  JOSEPH  PETZVAL,  in  his  celebrated  paper, 
Bericht  ueber  die  Ergebnisse  einiger  dioptrischer  Untersuchungen  (Pesth, 
1843.  Verlag  von  C.  A.  HARTLEBEN).1  The  formula  is  applicable 
only  in  case  the  image  is  stigmatic,  and  although  PETZVAL  does  not 
expressly  even  allude  to  this  pre-requisite  condition,  it  is  hardly  to 
be  supposed  that  he  was  ignorant  of  it.2 

1  See  also  J.  PETZVAL:  Bericht  ueber  optische  Untersuchungen.     Sitzungsber.  der  math.- 
naturwiss.  Cl.  der  kaiserl.  Akad.  der  Wissenschaften,  Wien,  xxvi  (1857),  50-75,  92-105, 
129-145.     The  PETZVAL-formula  is  given  here  also  without  proof,  on  p.  95,  but  the  re- 
mainder of  this  contribution  is  chiefly  devoted  to  a  discussion  of  this  equation,  which  is 
shown  to  hold  for  a  number  of  simple  special  cases. 

2  In  regard  to  this  question,  see  especially  M.  VON  ROHR'S  Theorie  und  Geschichte  des 
photographischen  Objektivs  (Berlin,  1899),  p.  270.     L.  SEIDEL,  in  his  paper,  "Zur  Dioptrik. 


440  Geometrical  Optics,  Chapter  XII.  [  §  304. 

The  conditions  that  an  optical  apparatus  consisting  of  a  centered 
system  of  spherical  refracting  surfaces,  provided  with  a  narrow  stop 
to  limit  the  widths  of  the  bundles  of  effective  rays,  shall,  as  a  first 
approximation,  produce  a  stigmatic  plane  image  of  a  plane  object, 
are  the  following: 


304.    Formulae  for  the  Magnitudes  of  the  Aberration-Lines. 

Assuming  that  we  have  a  plane  object,  we  obtain,  by  means  of 
formulae  (314)  and  (322),  the  following  expressions  for  the  magnitudes 
of  the  aberration-lines,  in  the  GAUssian  image-plane  <rm,  of  the  meridian 
and  sagittal  rays  of  a  narrow  astigmatic  bundle  of  image-rays: 


P'V  =  _*^ 

•*•      Ml.    '     »J. 


ni '  m 


(328) 


In  order  to  be  able  to  compare  these  formulae  with  SEIDEL'S  general 
formulae,  to  be  derived  hereafter,  we  shall  transform  them  by  the  aid 
of  several  approximate  relations,  which  may  be  introduced  here  with- 
out neglecting  the  magnitudes  of  the  3rd  order  of  smallness. 

Since  the  I.  and  II.  image-points  of  the  infinitely  narrow  bundle  of 
image  rays,  which  are  designated  by  S'm  and  3^,  respectively,  are  here 
supposed  to  be  not  very  far  from  the  axial  image-point  M'm,  we  may 
put 


and,  thus,  without  neglecting  magnitudes  of  the  3rd  order,  we  may 
write  here  the  following  formulae: 


where  the  symbols  u,  y,  z  have  the  same  meanings  as  in  §255. 
Moreover,  in  connection  with  these  equations,  we  may  employ  here 
the  Law  of  ROBERT  SMITH  (§  194),  and  write,  therefore,  according  to 

Ueber  die  Entwicklung  der  Glieder  ster  Ordnung,  welche"  u.  s.  w.,  Astr.  Nachr.,  xliii 
(1856),  Nos.  1027,  1028  and  1029,  pointed  out  (see  No.  1029)  that  the  PETZVAL-Equation 
implied  the  abolition  of  astigmatism;  as  was  remarked,  likewise,  by  H.  ZINKEN  gen. 
SOMMER  in  a  paper  entitled,  Ueber  die  Berechnung  der  Bildkruemmung  bei  optischen 
Apparaten,  POGG.  Ann.,  cxxii.  (1864),  563-574. 


§  304.]  Theory  of  Spherical  Aberrations.  441 

formulae  (92)  of  Chap.  V: 

nmhmy'm  _  njirfj       n'mhmz'm  _  n1A1zl 

Finally,  also,  by  formula  (155)  of  Chapter  VIII,  we  have: 


and,  thus,  we  obtain: 

dx» = i  •  ^' •  wT^^"  ^"«£'*r*«^'%%>  (330) 

If,  therefore,  employing  the  relation: 

we  take  from  under  the  two  summation-signs  in  each  of  the  formulae 
(328)  the  term 

and  if  we  multiply  both  sides  of  these  equations  by  nm/u'm,  at  the 
same  time  eliminating  d\'m  and  d\'m  on  the  right-hand  sides  of  the  two 
equations  by  means  of  the  formulae  (330),  and  also  expressing  y'm  in 
terms  of  yl  by  means  of  SMITH'S  Formula: 


we  obtain,  finally,  the  formulae  (328)  in  the  following  forms: 

(330 

--** 

where,  for  brevity,  we  put : 


442  Geometrical  Optics,  Chapter  XII.  [  §  305. 

ART.  98.     SPECIAL    CASES. 

305.     Case  of  a  Single  Spherical  Refracting  Surface. 

The  relations  between  the  curvatures  of  the  image-surfaces  and  the 
curvature  of  the  object-surface  are  given,  in  the  case  of  a  single  spherical 
refracting  surface,  by  formulae  (321).  For  a  plane  object  (R  =  oo), 
these  formulae  may  be  written  as  follows  : 

_!_       _  n'  -  n  _       tu\u-r)*(    i         _i\ 
R'  ~          nr          3H  r(u  -  u)\n'u'      nu)' 
i  n'  —  n        fu2(u  —  r}1  f    i          I  \ 

£'  =       ~nr         n  f*(«  -  f^VnV  ~  nu)  ' 

In  each  of  the  three  following  cases  we  shall  have  a  stigmatic  image 
whose  curvature  will  be  : 


nr 

(1)  When  u  —  o,  which  is  a  case  that  possesses  no  practical  interest; 

(2)  When  nu  =  n'u'  ,  in  which  case  the  conjugate  axial  points  M,  M' 
coincide  with  the  aplanatic  points  Z,  Z'  of  the  spherical  refracting 
surface  (§  207).     Under  these  circumstances,  it  does  not  matter  where 
the  stop  is  placed.     And,  finally: 

(3)  When  u  =  r\  that  is,  when  the  centre  0  (or  L)  of  the  stop  coin- 
cides with  the  centre   C  of  the  spherical  surface.     In  this  case  the 
chief  rays  proceed  in  straight  lines  from  the  points  in  the  plane  object 
to  the  conjugate  points  in  the  image. 

If  the  stop-centre  is  situated  at  the  vertex  A  of  the  spherical  surf- 
ace (u  =  o),  the  curvatures  of  the  two  image-surfaces  are: 


i  n'  —  n        ,  /    i          i  \ 

R'  nr  \n'uf      nu)' 


Lastly,  in  case  the  object  is  at  infinity  (u  =  oo),  the  curvatures  of 
the  image-surfaces  are: 

JL       _nr  -n\i        3  (u~ 
R'  '  r        *  ~*V        r 


i  it-nil       i  (u-  r\\ 

=  —  ™i«+>v—  J  i- 


30(5-]  Theory  of  Spherical  Aberrations.  443 

306.     Case  of  an  Infinitely  Thin  Lens. 

For  the  case  of  an  Infinitely  Thin  Lens,  we  can  write  : 

J 


wherein,  employing  the  same  special  Lens-Notation  as  in  §  268,  we 
may  put: 

r  r  r       (n  —  i)(c  —  x)  —  HP 

Jl  =  c-x,     Jl  =  c-x,     J2  =  ~  -. 

Introducing  these  symbols,  we  shall  find: 


where  the  symbol   U  is  used   as  an  abbreviation  for  the  following 
function  : 


2(n  +  i)             2n  n  +  i 

• — xx  + xv  +  - 


Thus,  the  curvatures  of  the  images  produced  by  a  Thin  Lens  will  be 
for  the  case  of  a  plane  object: 

U 


(i)  When  the  centre  of  the  stop  coincides  with  the  centre  of  the 
Infinitely  Thin  Lens  (x  =  oo),  we  find  U/(x  —  x)z  —  i,  and  hence: 

3^  +  1          i  n  +  1 


whence  it  appears,  that  under  such  circumstances,  the  curvatures  of 
the  image-surfaces  are  independent  of  the  distance  of  the  object  from 
the  Lens,  and  the  chief  rays  proceed  in  straight  lines  from  the  points 
of  the  object  to  the  conjugate  points  of  the  image.  The  curvatures, 
'in  fact,  depend  only  on  the  focal  length  of  the  Lens  and  the  value  of 
the  relative  index  of  refraction  (n),  but  not  on  the  form  of  the  Lens. 
If  n  =  3/2,  we  find  R'  =  -  3//n  and  R'  =  -  3//5. 
(2)  The  condition  of  the  stigmatic  image  is 

U  =  o, 


444  Geometrical  Optics,  Chapter  XII.  [  §  307, 

in  which  case  the  curvature  of  the  image  is: 

ii  <p 

R7  =  W  =    ~  n* 

(3)  In  the  special  case  of  a  System  of  Infinitely  Thin  Lenses  in 
Contact,  with  the  centre  of  the  stop  situated  at  the  common  vertex  of  the 
Lenses  (x  =  oo  for  each  Lens),  the  function  U'/(x  —  x)2  is  equal  to 
unity  for  each  Lens,  and,  hence,  the  curvatures  of  the  image-surfaces 
will  be: 


Accordingly,  the  condition  of  a  flat  stigmatic  image  in  the  neighbour- 
hood of  the  axis  (R'  =  R'  =  oo)  requires  that  we  shall  have  in  this 
case: 

S<p  =  o, 

which  means  that  the  combination  of  Lenses  must  act  like  a  slab  with 
plane  parallel  faces. 

VI.     ABERRATIONS  IN  THE  CASE  OF  IMAGERY  BY  BUNDLES  OF  RAYS  OF  FINITE  SLOPES 
AND  OF  SMALL  FINITE  APERTURES. 

ART.   99.     COMA. 

307.  The  Coma-  Aberrations  in  General.  Heretofore,  in  the  in- 
vestigations of  the  aberrations  in  the  case  of  object-points  not  on  the 
optical  axis,  it  has  been  assumed  always  that  the  rays  were  limited  by 
a  stop  of  infinitely  narrow  dimensions.  In  actual  optical  construction 
this  condition  can  never,  of  course,  be  absolutely  realized;  nor,  indeed, 
in  the  case  of  certain  optical  instruments  is  it  necessary  that  it  should 
be,  so  long  as  the  diameter  of  the  stop  is  relatively  very  small.  On 
the  other  hand,  when  it  is  required  to  produce  the  image  of  a  fairly 
extensive  object  by  means  of  somewhat  wide-angled  bundles  of  rays, 
as,  for  example,  is  often  the  case  with  photographic  objectives,  the 
diameter  of  the  stop  will  enter  as  a  chief  factor  in  the  study  of  the  aber- 
rations of  the  rays.  Thus,  whereas  we  saw  (§  304)  that  the  aberra- 
tion-lines in  the  case  of  infinitely  narrow  bundles  of  astigmatic  rays 
were  proportional  to  the  first  powers  of  the  aperture-co-ordinates  ylt 
z\  (§  259)  »  we  must  now  advance  a  step  farther,  and  assume  here  that 
the  aperture  is  so  wide  that  we  will  not  be  justified  in  leaving  out 
of  account  the  second  powers  and  products  of  these  co-ordinates. 

A  bundle  of  rays  of  finite  aperture,  emanating  from  a  point  outside 


307.] 


Theory  of  Spherical  Aberrations. 


445 


the  optical  axis,  may  show  aberrations  of  a  character  similar  to  the 
spherical  aberration  along  the  axis  of  a  direct  bundle  of  rays  (see  §  208 
and  §  260).  These  aberrations  will  be  manifest  in  both  the  meridian 
and  sagittal  sections  of  the  bundles  of  rays,  but  here  a  very  impor- 
tant difference  is  to  be  remarked,  as  will  now  be  explained. 

The  rays  of  the  sagittal  section  are  symmetrically  situated  on  op- 
posite sides  of  the  meridian  plane,  so  that  the  point  of  intersection  of 
every  pair  of  symmetrical  rays  in  this  section  will  lie  in  the  plane  of 
the  meridian  section,  for  example,  as  shown  in  Fig.  144.  But  in  the 


FIG.  144. 

SYMMETRICAL  CHARACTER  OP  THE  ABER- 
RATIONS OF  THE  RAYS  OF  THE  SAGITTAL 
SECTION  OF  AN  INCLINED  BUNDLE  OF  RAYS 
OF  FINITE  APERTURE.  The  chief  ray  of  the 
bundle  is  the  ray  marked  «.  The  plane  of 
the  meridian  section  is  the  plane  containing 
«  which  is  perpendicular  to  the  plane  of  the 
paper. 


FIG. 145. 

UNSYMMETRICAL  CHARACTER  OF  THE 
ABERRATIONS  OF  THE  RAYS  OF  THE  ME- 
RIDIAN SECTION  OF  AN  INCLINED  BUNDLE  OF 
FINITE  APERTURE.  The  chief  ray  of  the 
bundle  is  the  ray  marked  u.  This  is  the  ray 
which  at  some  stage  of  its  progress  goes 
through  the  centre  of  the  stop.  The  rays  of 
the  meridian  section  are  in  general  not 
symmetrical  with  respect  to  the  chief  ray. 


meridian  section  (Fig.  145)  it  is  obvious  that,  in  general,  there  will  be 
no  symmetry  at  all.  The  chief  ray  of  the  bundle  will  depend  on  the 
position  on  the  optical  axis  of  the  centre  of  the  stop.  If  the  rays  are 
received  on  a  screen  placed  perpendicularly  to  the  optical  axis,  and 
if  a  straight  radial  line  is  drawn  in  the  plane  of  the  screen  through 
the  point  where  the  screen  meets  the  optical  axis  and  intersecting  the 
light-pattern  on  the  screen,  there  will  be  no  symmetry  in  the  pencil 
of  rays  which  meet  the  screen  at  points  lying  along  this  line :  whereas 
in  the  case  of  a  pencil  of  rays  which  meet  the  screen  at  points  lying 
along  a  line  at  right  angles  to  this  radial  line  there  will  be  symmetry. 
The  light-pattern  on  the  screen  sometimes  presents  the  appearance  of 
a  comet,  with  its  tail  turned  either  towards  or  away  from  the  optical 
axis;  which  accounts  for  the  origin  of  the  name  "coma" '. 

So  far  as  the  meridian  rays  are  concerned,  we  have  to  ascertain  only 
the  ^-aberrations  (§  256),  because,  by  the  Laws  of  Refraction,  the  paths 

1  Some  excellent  drawings  exhibiting  these  appearances  are  to  be  found  in  H.  DENNIS 
TAYLOR'S  A  System  of  Applied  Optics  (London,  1906).  This  work  contains  several  chap- 
ters in  regard  to  Coma.  Especially  interesting  in  the  diagrams  are  the  drawings  by  Prof. 
S.  P.  THOMPSON,  Plate  XVI. 


446  Geometrical  Optics,  Chapter  XII.  [  §  307. 

of  these  rays  throughout  their  progress  through  the  centered  system 
of  spherical  surfaces  will  lie  wholly  in  the  meridian  or  :ry-plane,  so  that 
their  s-aberrations  will  all  be  equal  to  zero.  But  if  the  path'  of  the 
ray  lies  outside  of  this  plane,  we  shall  have  to  determine  its  s-aberration 
as  well  as  its  ^-aberration.  In  general,  the  ^-aberration  of  any  ray 
of  a  bundle  of  rays  may  be  considered  as  compounded  by  summa- 
tion of  the  ^-aberrations  of  the  meridian  rays  and  of  the  sagittal 
rays. 

Evidently,  in  the  case  of  a  pair  of  rays  of  the  sagittal  section  which 
are  symmetrically  situated  on  opposite  sides  of  the  meridian  plane  the 
values  of  £  for  the  two  points  where  these  rays  cross  the  plane  of  the 
Entrance-Pupil  (§  257)  will  be  equal  in  magnitude  but  opposite  in  sign; 
and,  hence,  the  position  of  the  point  in  the  meridian  plane  where  these 
two  rays  meet  after  traversing  the  optical  system  must  be  independent 
of  the  sign  of  £.  If  one  of  these  rays  crosses  the  GAUssian  image- 
plane  a'  in  a  point  whose  co-ordinates  are  given  by  77',  f ',  the  other  ray 
will  cross  this  plane  at  the  point  ??',  —  £';  and,  hence,  T\ ',  and,  there- 
fore, also,  the  ^-aberration  by',  will  be  independent  of  the  sign  of  £ 
(or  of  z) .  Accordingly,  in  the  series-development  of  the  ^-aberration  of 
a  ray  belonging  to  the  sagittal  section,  there  can  be  no  term  involving 
the  odd  powers  of  the  co-ordinate  z;  and,  as  we  propose  to  consider 
here  no  terms  involving  powers  of  the  aperture-co-ordinates  y,  z  above 
the  second,  obviously,  the  only  terms  that  can  occur  in  the  series- 
developments  of  the  ^-aberrations  will  be  terms  involving  y2  and  z2 

(^§259). 

When  we  come  to  consider  the  z-aberration,  we  find  that  the  case 
is  exactly  opposite  to  that  of  the  ^-aberration ;  for,  since  the  aberration 
5z'  changes  sign  along  with  change  of  the  sign  of  z,  the  series-develop- 
ment of  dz'  can  contain  terms  which  involve  only  the  odd  powers  of 
the  aperture-co-ordinate  z;  so  that,  within  the  limits  prescribed  for 
the  present  investigation,  the  only  term  in  the  series-development  of 
dz'  will  be  the  term  involving  the  product  yz. 

The  complete  investigation  of  these  so-called  "Comatic"  Aberrations 
is  quite  laborious.  We  shall  consider  here  only  the  ^-aberration  of  a 
ray  lying  in  the  meridian  plane,  the  series-development  of  which  will 
contain  only  the  term  involving  y2.  The  reader  who  is  interested  in 
the  investigation  of  the  y-  and  z-aberrations  of  a  ray  belonging  to 
the  sagittal  section  of  the  bundle  of  rays  will  find  the  whole  subject 
exhaustively  treated  by  KOENIG  and  VON  RoHR.1 

1  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen:  Chapter 
V  of  Volume  I  of  Die  Theorie  der  optischen  Inslrumente  (Berlin,  1904),  edited  by  M.  VON 
ROHR;  see  pages  265-292. 


308.] 


Theory  of  Spherical  Aberrations. 


447 


308.  The  Lack  of  Symmetry  of  a  Pencil  of  Meridian  Rays  of 
Finite  Aperture.  In  the  special  case  when  the  chief  ray  of  the  bundle 
coincides  with  the  optical  axis,  there  will  be  symmetry  in  the  pencil 
of  meridian  rays,  as  is  exhibited  in  the  diagram  (Fig.  146),  which 
represents  the  meridian  section  of  an  optical  system  consisting  of  a 
single  spherical  surface.  The  centre  of  the  stop  is  supposed  here  to 


FIG.  146. 


I«ACK  OF  SYMMETRY  OF  A  PENCIL,  OF  MERIDIAN  RAYS  OF  FINITE  APERTURE. 

be  situated  at  the  vertex  A  of  the  spherical  surface,  and  the  object 
is  infinitely  distant,  so  that  the  object-rays  emanating  from  any  point 
of  the  object  are  parallel. 

If  the  object-point  is  not  on  the  optical  axis,  the  chief  ray  of  the 
bundle  of  object-rays  will  be  inclined  to  the  optical  axis  at  some 
angle,  say  6;  and  it  is  evident  by  an  inspection  of  the  figure  that  the 
meridian  rays  of  this  bundle  produce  an  effect  quite  different  from  that 
which  we  perceived  in  the  case  of  a  bundle  of  rays  emanating  from  an 
axial  object-point.  In  the  first  place,  the  chief  ray  is  no  longer  the 
ray  which  meets  the  spherical  refracting  surface  normally;  and, 
generally,  this  will  always  be  a  distinguishing  peculiarity  of  such  a 
pencil  of  meridian  rays,  so  that  the  chief  ray  will  not  (except  for  certain 
special  positions  of  the  stop)  go  through  the  centre  C  of  the  spherical 
surface;  and  even  in  case  it  did  happen  to  pass  through  the  centre  of 
one  surface,  it  would  not  pass  through  the  centre  of  the  next  following 
surface  of  a  centered  system  of  spherical  surfaces.  The  straight  line 
drawn  through  C  parallel  to  the  incident  rays  (which  may,  or  may  not, 
be  the  path  of  an  actual  ray  of  the  pencil) ,  is  in  a  certain  sense,  an  axis 
of  symmetry  for  the  refracted  rays  in  the  same  way  as  the  optical 
axis  is  an  axis  of  symmetry  for  the  direct  pencil  of  refracted  meridian 
rays:  but,  since  the  stop  cuts  off  more  rays  on  one  side  of  this  line  than 
it  does  on  the  other,  the  actual  pencil  of  refracted  rays  is  not  symmet- 
rical with  respect  to  this  straight  line  of  slope-angle  6  drawn  through 


448 


Geometrical  Optics,  Chapter  XII. 


[  §  309. 


the  centre  C  of  the  spherical  surface.  Almost  exactly  the  same  un- 
symmetrical  effect  would  be  obtained  with  the  direct  pencil  of  meridian 
rays  if  the  centre  of  the  stop,  instead  of  lying  on  the  optical  axis, 
were  situated  above  or  below  the  axis.  In  fact,  if  the  diameter  of 
the  stop  is  increased  in  the  ratio  I  :  cos  8,  and  if  at  the  same  time  the 
centre  of  the  stop  is  displaced  at  right  angles  to  the  axis  by  an  amount 
equal  to  r-sin  8,  we  shall  obtain  precisely  the  same  character  of  effect 
with  the  direct  pencil  of  rays  as  is  obtained  with  the  inclined  pencil 
in  the  case  shown  in  the  figure. 

In  general,  therefore,  we  can  say  that  the  image  of  a  point  outside 
the  axis  produced  by  a  wide-angle  pencil  of  meridian  rays  will  never 
be  a  point,  but  a  piece  of  a  caustic  curve  formed  by  the  I.  Image- 
Points  of  the  succession  of  infinitely  narrow  pencils  of  which  the  entire 
finite  pencil  may  be  supposed  to  consist. 


ART.  100.     FORMULAE  FOR  THE  COMATIC  ABERRATION-LI JJES. 

309.  Invariant-Method  of  Abbe.  In  order  to  ascertain  the  nature 
of  an  element  of  this  caustic  curve,  we  shall  employ  the  method  of 
ABBE,  as  given  both  by  CzAPSKi1  and  by  KOENIG  and  VON  RoHR.2 

In  Fig.  147  the  plane  of  the 
paper  represents  the  plane 
of  the  meridian  section  of 
the  bundle  of  rays ;  and  the 
letters  C  and  A  designate 
the  centre  and  vertex,  re- 
spectively, of  one  of  the 
surfaces  of  the  centered 
system  of  spherical  surfaces 
(AC  =  r).  The  points  B 
and  /  lying  in  the  me- 
ridian section  of  the  surf- 

/c/^a'+rfa',  ZZ?T"/=<A',  arc*/=/.  ace    are     two     incidence- 

points  very  near  together. 

BS'  and  IR'  represent  the  paths  of  two  refracted  meridian  rays 
corresponding  to  two  incident  meridian  rays  SB  and  RI,  respectively 
(which  latter,  however,  are  not  shown  in  the  diagram).  The  points 
S'  and  R'  designate  the  positions  on  BS'  and  IR'  of  the  I.  Image- 

1  S.   CZAPSKI:   Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,   1893),  pages 
115-118. 

2  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen:  Chapter 
V  of  Vol.  I  of  Die  Theorie  der  optischen  Instrumente  (Berlin,  1904),  pages  270-273. 


w" 


FIG.  147. 

COMATIC  ABERRATIONS  S'7*  AND  S' W  OF   AN  IN- 
FINITELY NARROW  PENCIL  OF  MERIDIAN  RAYS. 


§  309.]  Theory  of  Spherical  Aberrations.  449 

Points  corresponding  to  the  points  5  and  R  on  the  incident  rays  SB 
and  RI,  respectively  ;  the  actual  positions  of  S'  and  Rr  being,  of  course, 
dependent  on  the  positions  of  5  and  R,  respectively.  The  angles  of 
incidence  at  B  and  /  are  supposed  to  differ  from  each  other  by  an  infi- 
nitely small  magnitude  of  the  1st  order;  and,  consequently,  the  points 
designated  by  S'  and  R'  are  two  infinitely  near  points  on  the  caustic 
curve  of  the  meridian  rays.  The  point  of  intersection  of  the  refracted 
rays  BSf  and  IR'  is  designated  in  the  figure  by  T'\  and  we  may  con- 
sider S'T'  as  the  longitudinal  aberration  along  BS'  of  the  infinitely 
narrow  pencil  of  meridian  rays  which  are  refracted  at  the  points  lying 
in  the  arc  BI. 

The  following  symbols  may  be  conveniently  employed  : 

Z  CBS'  =  a',     Z  CIT'  =  a'  +  da',     /  ICB  =  d<p,     Z  BTI  =  d\', 
BS'  =  s',    IR'  =  s'  +  ds'. 

With  T'  as  centre  and  with  radii  equal  to  T'l  and  T'R',  describe 
two  circular  arcs  meeting  BS'  in  the  points  designated  in  the  figure 
by  Y'  and  Z',  respectively.  The  variation  ds'  =  IR'  —  BS'  may  be 
considered  as  consisting  of  a  displacement  S'Z'  together  with  a  dis- 
placement Z'R'.  The  latter  may  be  said,  in  a  certain  sense,  to  be 
due  to  the  variation  of  the  point  of  incidence  from  B  to  /;  whereas 
the  former  is  the  displacement  depending  on  the  angle  d\'  between 
the  refracted  rays  leaving  B  and  /.  We  shall  try  now  to  obtain  an 
expression  for  the  magnitude  of  the  component 

S'Z'  =  dq' 

of  the  total  variation;  because,  since  BT'  and  IT'  are  tangents  to  the 
caustic  curve  at  the  two  infinitely  near  points  S'  and  R',  and  since, 
therefore,  the  lengths  S'T'  and  T'R'  can  differ  from  each  other  only 
by  an  infinitesimal  magnitude  of  an  order  higher  than  either  of  them, 
so  that  we  can  put 

ST  =  T'R'  = 


the  magnitude  denoted  by  dq'  is  equal  to  twice  the  aberration  S'T'. 

Incidentally,  also,  we  may  observe  that  since  (neglecting  infinitesi- 
mals of  the  2nd  order)  S'T'  +  T'R'  =  dq'  =  the  length  of  the  element 
of  the  caustic,  the  radius  of  curvature  of  the  caustic  at  '5'  is  equal  to 
dq'/d\'. 

Throughout  this  present  investigation  we  shall  retain  magnitudes 
of  the  order  dtp.  Hence,  provided  we  neglect  only  small  magnitudes  of 
an  order  higher  than  the  ist,  we  shall  obtain  from  the  figure  the 

30 


450  Geometrical  Optics,  Chapter  XII.  [  §  309. 

following  useful  relations: 

IV  =  -  r-cosa'-^; 
also, 

IY'  =  Y'T'-dK'  =  s'-d\'; 
and,  hence, 

d\'          r-  cos  a' 
^  =         ~' * 
Moreover, 

da'  =  d\f  +  d<p,     BY'  =  r-sin  a'-d<p. 
Now 

ds'  =  IR'  -  BS'  =  Y'Z'  -  BS'  =  S'Z'  -  BY'; 
that  is, 

ds'  =  dq'  -  BY'; 
and,  since 

d£_d^_<ti^          r-cosa'  dq' 
d<p  ~  d\'  d<p  =  s'       d\' ' 

we  obtain: 

ds'          r  •  cos  a'  dq' 

—  = -t —  -Trr  —  r-sm  a  . 

dp  s'       d\' 

In  order  to  obtain  now  an  expression  for  ds'/dp,  we  must  employ 
the  Law  of  Refraction,  which  ABBE  does  by  introducing  here  the 
invariant-function  of  astigmatic  refraction  (§299),  viz.: 

COStt\  /I         COSd' 


5 
Accordingly,  differentiating  Q  with  respect  to  <p,  we  obtain : 

dO  ./i      2cosa'Wa'    ,  n'-cos2a'  ds' 

— -  =  —  n  •  sin  a  I • -, —  1 -j—  H ^—  3— ; 

a<p  \r  s       /  d(p  s         d<p 

wherein  let  us  put : 

da'  d\'  r  •  cos  a' 

dtp  d(f>  s' 

and  let  us,  also,  substitute  for  ds'/d<p  the  expression  which  we  obtained 
above  in  terms  of  dq'/d\'.  Thus,  after  several  simple  transformations, 
we  derive  the  following  equation : 

I  dQ  -      ^  •  3^'Q      it'- cos*  of  dq' 
rd^>~    ~7^   n's'          ~^       d\" 
where 

K  —  n  -  sin  a  =  n'  -  sin  a' 

denotes  the  so-called  "optical  invariant". 


§  310.]  Theory  of  Spherical  Aberrations.  451 

The  above  formula  has  been  derived  for  the  rays  after  refraction 
at  the  spherical  surface  here  considered  ;  but  it  is  obvious  that  we  shall 
obtain  in  the  same  way  a  precisely  similar  relation  connecting  the 
corresponding  magnitudes  before  refraction,  viz.  : 

l^Q-  _  K  i     K'  -3 

~    ~   2 


ns  s3       d\' 

Combining,  therefore,  these  two  formulae,  and  using  ABBE'S  difference- 
notation,  we  obtain: 

Oidq 


Thus,  knowing  the  values  of  the  magnitudes  denoted  by  a,  s,  dq  and 
d\,  which  relate  to  the  narrow  pencil  of  meridian  rays  before  refraction 
at  the  spherical  surface,  we  can  calculate  the  magnitudes  denoted  by 
a'  and  s',  and  determine,  by  means  of  the  formula  just  obtained,  the 
magnitude  of  the  ratio  dy'/d\',  which  relates  to  the  pencil  of  rays 
after  refraction. 

310.  Instead  of  a  single  spherical  surface,  let  us  suppose  now  that 
the  optical  system  consists  of  m  spherical  surfaces  with  their  centres 
ranged  all  along  one  straight  line.  Introducing  in  our  notation  the 
surface-subscripts,  we  must  write  : 


and,  hence,  for  a  centered  system  of  m  spherical  surfaces,  we  obtain 
by  formula  (333)  the  following  recurrent  formula: 


n'm  -  cos3  am  dqm          dql  (  s(  •  s'2  •  •  •  s'm_l  \s  /  cos  c^  •  cos  a2  •  •  •  cos  am  V 
s'^3        d\'m        1  d\  \  sl  •  s2  •  -  •  sm  )  \  cos  ai  •  cos  a'2  •  •  •  cos  a^_x  / 
g /V&r  •  •^-AVcosa?+..cosa4+2.  •  •cosaE.V  /  L\ 

fet  \^+r^+2- '  '*W/  Vcosai-cosa^!-  •  -cosa^/  Vw^ 

If  we  write 

then 

t  Sfc '  Ci'\j1          Sje '  d^jf 

Ju  cos  ak          cos  %' 
and,  hence: 

U         s',,     cos 


J*+i      ^*+i     cos  *k 


452  Geometrical  Optics,  Chapter  XII.  [  §  311. 

Therefore, 

Ji  __  Ji'Jt' '  'Jm-i  _  V  V  '  's'm-i     cos  a2'cos  0.3*  '  -cos  gm 
7™         J9'j*'''jm         S2'ss'''sm     cos  ai -cos  a2- • -cos  an)_t* 

j ni  »/ A    */o  J m  &       3  *  wi     i 

Thus,  the  recurrent  formula  obtained  above  may  be  put  in  the  fol- 
lowing form : 

C   dq'm_   ,.  /"JiVcos3^   ^ 

c/Xi 


;     (334) 

If,  as  is  usual,  the  bundle  of  object-rays  is  homocentric  (dg^  =  o), 
the  formula  above  may  be  written  as  follows: 


311.     If  a  screen.  is  placed  perpendicularly  to  BS'  at  5',  the  pencil 
of  meridian  rays  will  intersect  this  screen  in  the  aberration-line 

S'W  =  dw'\ 
where 


denotes  a  magnitude  of  the  second  order  of  smallness  as  compared 
with  d\r.    Hence, 

n'-cossa'd'       2 


Accordingly,  by  formula  (333),  for  the  &th  spherical  surface  we  have: 


The  product  nk'd\'k  is  the  so-called  "numerical  aperture"  (cf.  §  364)  of 
the  pencil  of  rays  after  refraction  at  the  &th  spherical  surface,  and 
dw'k  here  is  analogous  to  the  Lateral  Aberration  in  the  case  of  a  direct 
bundle  of  rays  (see  §  262  and  §  266). 

If  we  give  k  in  succession  all  integral  values  from  k  =  I  to  k  =  m, 
and  put  dwl  =  o,  we  obtain,  by  addition: 


§  312.]  Theory  of  Spherical  Aberrations.  453 

and  if  here  we  substitute  : 

fci     and    X 


we  can  write  finally: 

w;^=_3  .i(  2          J  -  /£Y(*Ya.*A.A(jL)  .  (336) 

sm          2   jm^  cos2al'cosam^[  \jj  v  \nsjk  w  ' 

312.  Let  us  now  impose  the  condition  that  the  slope-angles  6,  6'  of 
the  chief  rays  are  small  magnitudes  of  the  first  order  —  of  the  same  order 
as  the  aperture-angles  X,  X',  as  we  shall  now  denote  these  latter  angles, 
instead  of  denoting  them,  as  above,  by  the  symbols  d\,  d\r.  Without 
neglecting  ultimately  the  magnitudes  of  the  3rd  order  of  smallness, 
we  may  obviously  introduce  in  the  above  formula  (336)  the  approxi- 
mate values  of  the  magnitudes  denoted  by  the  symbols  s,  j,  Q  and  K, 
Thus,  we  may  employ  here  the  approximate  relations: 

cos  a  =  i,     sin  a  =  a,     6  .=  —  hju    and     <(>  =  h/r; 

where  h  denotes  the  incidence-height  of  the  chief  ray  and  u  =  AM. 
And,  hence,  since 

a  =  6  +  <)>, 
we  can  put: 

hJ 

a  =  ~n~> 
and,  therefore: 

K  =  n  •  sin  a  =  no.  =  hJ. 

Moreover,  approximately,  also: 

s  =  «, 

and,  hence,  if  h  denotes  the  incidence-height  of  a  paraxial  object-ray 
emanating  from  the  axial  object-point  Mlt  we  may  use  here  also  the 
following  relation: 


Finally,  we  may  put  here  Q  =  J.  Accordingly,  introducing  these 
values  in  formula  (336),  and  at  the  same  time  writing  now  5wf  in  place 
of  dw',  we  obtain: 


454 


Geometrical  Optics,  Chapter  XII. 


§313. 


313.  If  the  focussing-screen  is  placed  perpendicularly  to  the  chief 
image-ray  US'  (Fig.  148),  not  at  the  I.  Image-Point  S',  but  at  some 
other  point  S",  the  Lateral  Aberration  of  the  meridian  rays  will  now 
be  S"W"  and  from  the  diagram  we  obtain: 


S'W 


S"S' 
S'T' 


Now  if  the  screen  in  its  new  position  has  been  displaced  so  little  with 

respect  to  its  first  position 
that  S"Sr  is  of  the  same  order 
of  smallness  as  S'W',  that  is, 
if  S"Sr  is  of  a  higher  order 
of  smallness  than  S'T',  we 
may  put 

S"W"  =  S'W. 


H" 


FIG.  148. 


COMATIC  ABERRATION  OF  MERIDIAN  RAYS  MEASURED 
IN  A  PLANE  PERPENDICULAR  TO  THE  OPTICAL,  AXIS. 


And,  moreover,  if  now  the 
focussing-screen  is  rotated 
about  an  axis  perpendicular 

to  the  plane  of  the  diagram  at  5"  until  it  is  perpendicular  to  the  optical 

axis  at  the  point  M",  then,  since 

we  have,  neglecting  magnitudes  of  orders  higher  than  0'  : 
S'W'- cos  LM"S"W"  =  S'W', 

and,  hence,  the  formula  (337)  derived  above  is  valid  also  in  case  the 
aberration-line  is  measured  in  a  transversal  plane  M"S"  at  right  angles 
to  the  optical  axis,  provided  this  plane  is  not  too  far  removed  from  the 
I.  Image-Point  S'.  In  particular,  the  formula  (337)  is  valid  if  the 
aberration-line  dw'  is  measured  in  the  GAUSsian  Image-Plane  a'  per- 
pendicular to  the  optical  axis  at  M',  since  the  distance  from  this  plane 
of  the  I.  Image-Point  S'  is  of  a  higher  order  of  smallness  than 

M'P'  =  i', 

which  is  of  the  same  order  as  0'. 

Finally,  if  we  introduce  the  approximate  relations: 


and 


§  315.]  Theory  of  Spherical  Aberrations.  455 

we  may  write  formula  (337)  in  the  following  form: 


a    i 


kh 

7-  I  y~  J  7, 


i  \ 

I   .   (3  3  o  ) 

*y» 


Thus,  on  the  assumption  that  the  slope-angles  of  the  chief  rays  are 
small  magnitudes,  the  condition  of  the  abolition  of  the  so-called 
"Comatic"  Aberration  of  the  meridian  rays  is: 

o.  (339) 

Moreover,  if  the  reader  will  investigate  also  the  y-aberration  and  the 
s-aberration  of  a  ray  of  the  sagittal  section,  as  is  done,  for  example, 
by  Messrs.  KOENIG  and  VON  ROHR/  he  will  discover  that  equation 
(339)  is  likewise  the  condition  of  the  abolition  of  both  aberrations  of 
the  sagittal  rays. 

It  will  be  recalled  that  precisely  this  same  equation  was  obtained 
also  as  the  expression  of  the  Sine-  Condition  (formula  304). 

ART.    101.     SPECIAL   CASES. 

314.  Case  of  Single  Spherical  Surface.    The  condition  that  the 
comatic  aberration,  in  the  case  of  a  single  spherical  refracting  surface, 
shall  vanish  is  evidently  : 

*  JJ(i/n'ur  -  i/nu)  =  o; 

which  will  be  satisfied  in  each  of  the  three  following  cases  : 

(1)  /  =  o,  or  u  =  uf  =  r:  that  is,  when  the  object  and  image  co- 
incide at  the  centre  of  the  spherical  surface  —  a  case  possessing  no 
practical  interest; 

(2)  J  =  o,  or  u  =  r:  that  is,  when  the  stop-centre  is  situated  at 
the  centre  of  the  spherical  surface;  and 

(3)  nu  =  n'u':  that  is,  when  the  pair  of  conjugate  axial  points 
M,  M'  are  the  aplanatic  pair  of  points  of  the  spherical  surface. 

315.  Case  of  Infinitely  Thin  Lens.     Employing  the  usual  special 
Lens-Notation  (see  §268),  we  may  write  the  expression  on  the  left- 
hand  side  of  formula  (339)  as  follows: 

VxCi/ww'i  -x)  +  J2J2(x'  ~  ifnu{)  =  *>F; 

1  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen:  Chapter 
V  of  Vol.  I  of  Die  Theorie  der  optischen  Instrumente  (Berlin,  1904);  edited  by  M.  VON 
ROHR.  See  pages  275-289. 


456  Geometrical  Optics,  Chapter  XII.  [  §  316. 

where 

/!  =  c  —  x,     Jl  =  c  —  *, 

(n-  i)(c-x)  ~rcy>  (rc 

J**  n-i  '     '/2- 


Thus,  we  find: 


The  value  of  V  will  be  a  minimum  when  : 
n(zn 

"" 

For  real  values  of  c,  we  must  have  : 

(n  +  i)(5»  +  i)    2      4*-*     2  .   (»  +  i)2 

2  -2*>-  2 


2 

--  <px 
* 


n 


VII.    SEIDEL'S  THEORY  OF  THE  SPHERICAL  ABERRATIONS  OF  THE  THIRD  ORDER. 

ART.  102.     DEVELOPMENT  OF  SEIDEL'S  FORMULAE  FOR  THE  v-  AND  *- 

ABERRATIONS. 

316.  Gaussian  Parameters  of  Incident  and  Refracted  Rays.  If 
we  take  the  vertex  A  of  the  spherical  refracting  surface  as  the  origin 
of  a  system  of  rectangular  axes,  and  choose  the  positive  direction  of  the 
optical  axis  as  the  positive  direction  of  the  jc-axis,  then,  adopting  the 
method  of  GAUSS,  l  we  can  write  the  equations  of  the  incident  ray  as 
follows  : 


where  the  two  pairs  of  constants  B,  P  and  C,  Q  are  the  four  param- 
eters which  are  used  here  to  determine  the  position  of  the  incident 
ray.  And,  similarly,  the  equations  of  the  corresponding  refracted 

1  C.  F.  GAUSS:  Dioptrische   Untersuchungen  (Goettingen,  1841),  page  3. 


§  316.]  Theory  of  Spherical  Aberrations.  457 

ray  may  be  written  as  follows: 


where  B',  Pf  and  C'  ,  Qf  denote  the  corresponding  parameters  of  the 
refracted  ray.  In  these  equations  n,  n'  denote  the  absolute  indices  of 
refraction  of  the  first  and  second  medium,  respectively.  The  relations 
between  the  parameters  of  the  incident  ray  and  those  of  the  refracted 
ray,  whereby,  knowing  the  former,  we  can  determine  the  latter,  are 
obtained  by  GAUSS  very  simply  as  follows  : 
The  abscissa  of  the  incidence-point  B  is: 

AD  =  r(i  -  cos  <p)  =  2r-sin2-, 

where  D  designates  the  foot  of  the  perpendicular  let  fall  from  B  on 
the  optical  axis,  and  where  r  =  A  C  denotes  the  abscissa  of  the  centre 
C  of  the  spherical  surface,  and  (p  =  Z  B  CA  denotes  the  central  angle. 
Since  the  point  B  is  common  to  both  the  incident  and  refracted  rays, 
the  value  x  =  r(i  —  cos  <p)  must  satisfy  both  sets  of  equations;  and, 
consequently,  we  obtain: 


C  <P  Cr  „     (D 

2  —  r  •  sin2  —  +  O  =  2  — r  r  •  sin2  -  -  +  0'. 

n  2        ^          n'  2         ^    J 


(340) 


Moreover,  let  H,  H'  designate  the  points  where  the  incident  and 
refracted  rays,  produced  if  necessary,  cross  the  transversal  plane  per- 
pendicular to  the  optical  axis  at  the  centre  C  of  the  spherical  surface. 
Since,  according  to  the  Laws  of  Refraction,  BHr  lies  in  the  plane 
containing  BH  and  BC,  the  three  points  C,  H  and  H'  must  lie  all 
in  a  straight  line:  and  if  in  the  triangles  BHC,  B  H'  C  the  angles  at 
H,  H'  are  denoted  by  n,  /*',  the  following  relation  can  easily  be  deduced 
(see  Chap.  IX,  formula  (209))  from  the  law  connecting  the  angles  of 
incidence  and  refraction: 

n-  CH-smn  =  «'•  Ctf'-sin/i'. 

Accordingly,  if  the  co-ordinates  of  H,  H'  are  (r,  yh,  zh),  (r,  y'hJ  %), 
respectively,  we  shall  have: 

y*-~  +  p>  s'.  =  ~  +  e 


458  Geometrical  Optics,  Chapter  XII.  [  §  317. 

and 

y't-^  +  r.  <-%  +  <?•. 

and  since 


yh~  zh~  CH  ""n'-sin/i" 
we  obtain: 

(Br  +  nP)  sin  p  =  (5V  +  w'P')  sin  //,  1 

(O  +  w<2)  sin  M  =  (C'r  +  «'(?')  sin  //.  J 

By  means  of  these  formulae  (340)  and  (341),  we  can  obtain  the  values 
of  the  parameters  B',  Pr  and  C,  Q'  of  the  refracted  ray  in  terms  of 
those  of  the  incident  ray.1 

317.  Approximate  Values  of  the  Gaussian  Parameters,  and  the 
Correction-Terms  of  the  3rd  Order.  In  the  following  investigation  it 
is  assumed  that  the  aperture  of  the  optical  system  is  relatively  small, 
so  that  none  of  the  effective  rays  are  very  far  from  the  optical  axis. 
This  being  the  case,  we  may  regard  the  parameters  denoted  here  by 
B,  P,  C,  Q  and  B',  P',  C',  Q'  as  being  all  small  magnitudes  of  the  first 
order.  For  the  same  reason,  the  magnitudes  sin  <p,  cos  ju,  cos  /*'  are 
likewise  to  be  considered  as  small  magnitudes  of  the  1st  order.  We 
propose,  according  to  L.  SEIDEL,2  to  neglect  here  all  terms  of  orders 
higher  than  the  3rd;  and,  hence,  if  A  denotes  a  small  magnitude  of 
the  first  order,  we  may  write  this  as  follows  : 

A  =  a  +  5a; 

where  the  small  letter  a  denotes  the  part  of  A  which  is  of  the  ist 
order,  and  da  denotes  the  correction-term  of  the  3rd  order;  for,  as 
was  explained  in  §  254,  if  the  parameters  of  the  ray  are  regarded  as 
magnitudes  of  the  ist  order,  the  series-developments  will  contain  only 
terms  of  the  odd  orders. 

If,  therefore,  in  the  exact  formulae  (340)  and  (341)  we  substitute 
for  B,  P,  etc.,  b  +  db,  p  +  5£,  etc.,  respectively,  we  shall  obtain  a 
set  of  approximate  formulae  which  are  accurate  except  for  residual 
errors  of  the  5th  and  higher  orders.  Moreover,  each  of  the  new  equa- 
tions thus  obtained  will  break  up  at  once  into  two  others,  since,  evi- 
dently, the  terms  of  the  ist  order  on  one  side  of  the  equation  must  be 

1  See  also  OSCAR  ROETHIG:  Die  Probleme  der  Brechung  und  Reflexion  (Leipzig,  1876), 
pages  15-26. 

2  L.  SEIDEL:  Zur  Dioptrik.    Ueber  die  Entwicklung  der  Glieder  3ter  Ordnung,  welche 
den  Weg  eines  ausserhalb  der  Ebene  der  Axe  gelegenen  Lichtstrahles  durch  ein  System 
brechenden   Medien,  bestimmen:    Astronomische    Nachrichten,  xliii.   (1856),   Nos.   1027, 
1028,   1029. 


§  318.] 


Theory  of  Spherical  Aberrations. 


459 


equal  to  the  terms  of  the  same  order  on  the  other  side;  and  since  the 
same  is  true  also  in  respect  to  the  terms  of  the  3  rd  order.  Thus  between 
the  approximate  values  b,  p,  etc.,  and  b',  p',  etc.,  of  the  parameters  of 
the  ray  before  and  after  refraction  we  obtain  the  following  set  of 
relations : 


(342) 


and  between  the  correction- terms  of  the  3rd  order  the  following  re- 
lations : 


(343) 


)-:  6  +  ~r    (cosV-cosV); 


Obviously,  in  the  further  development,  it  will  be  sufficient  to  obtain 
the  formulae  for  the  magnitudes  b,  p,  b',  p'  which  relate  to  the  xy- 
plane ;  then  all  we  shall  have  to  do  to  find  the  corresponding  formulae 
for  the  magnitudes  c,  q,  c',  q'  which  relate  to  the  #2-plane  will  be  to 
substitute  in  the  first  formulae  the  latter  magnitudes  in  place  of  the 
former. 

318.  Relations  between  the  Ray-Parameters  of  Gauss  and  Seidel. 
Instead  of  the  GAUSsian  parameters 

B  =  b  +  8b,     P  =  p  +  8p     and     C  =  c  +  8c,     Q  =  q  +  8q, 
we  have  now  to  introduce  the  parameters 

rj  =  y  +  8y,     f  =  z  +  dz    and     i\  =  y  +  8y,     £  =  z  +  8z, 

which  are  employed  by  SEIDEL(§255),  and  which  are  the  co-ordinates 
of  the  points  P,  P  where  the  ray  crosses  the  two  fixed  transversal 
planes  o-,  <r,  respectively.  The  abscissae  of  the  points  M ,  M  where  the 
optical  axis  meets  the  transversal  planes  o-,  <r  will  be  denoted  by  u,  u, 
respectively;  thus, 

AM  =  u,     AM  =  u; 

and,  similarly,  for  the  pair  of  axial  points  Mf,  M'  conjugate  to  M,  M, 
respectively,  let  us  put: 

AM'  =  u',     AM'  =  u'. 


460  Geometrical  Optics,  Chapter  XII. 

Moreover,  as  in  §  255, 


[§318. 


7          (l       l\         *(l        l\ 
J  =  n  I  -  —  -  )  =  n  I ,  I, 

\r      u)          \r      u' J1 

f          /I       i\         ,/i        i\ 

J  =  n  [- )  =  w'{  -  -  —  I . 

\r      uj          \r      u'J 


Then,  since  the  incident  ray  must  go  through  the  points  P(u,  17, 
and  P(u,  i],  £),  we  must  have: 


(J-J)     uu    ' 


and,  hence,  for  the  approximate  values  we  have  the  following  relations: 


n 


y-y 


J  —  J     uu 

.2 


n 


z  —  z 


Cx    ~^       -r  w  ,  W 7- 

J  —  J     uu  J  — 

and  for  the  correction- terms  of  the  3rd  order: 


(344) 


fry  -&y 


8z  —  8z 


n 


J  —  J 


uu 


by      Sy 

^ 

5z       dz 


(345) 


and  by  priming  all  the  letters  in  formulae  (344)  and  (345),  except  the 
letters  J,  /,  we  shall  obtain  also  the  corresponding  relations  for  the 
refracted  ray. 

Formulae  (342)  and  (343)  lead  to  the  following  invariant  relations 
between  the  approximate  values  of  the  parameters  of  the  incident  and 
refracted  rays: 

n'yf      ny      n'zf      nz 


u 


'zf 
' 


u 


u 


n'yf     ny      n'z'      nz 


u 


u 


(346) 


u         u 

which   will  be  recognized  as  equivalent  to  the  well-known  law  of 


§  319.]  Theory  of  Spherical  Aberrations.  461 

ROBERT  SMITH  for  a  single  spherical  refracting  surface  (Chap.  VIII, 

§  194). 
Moreover,  we  find: 


J  — 


J  —  J        u 


u 


and,  hence,  substituting  these  values  in  the  first  and  third  of  formulae 
(343),  we  obtain,  after  some  obvious  reductions: 


u  u  V          u  2          n        u 


Combining  these  two  equations  so  as  to  eliminate  the  difference 
A(n-dy/u),  we  find: 


319.  It  only  remains  now  to  obtain  expressions  for  the  small  magni- 
tudes v,  cos  M,  cos  //  ;  wherein,  however,  we  need  consider  only  the 
terms  of  the  ist  order,  since  these  alone  will  have  any  influence  of  the 
3rd  order  on  the  value  of  the  expression  for  A(n-dy/u). 

In  order  to  obtain  the  approximate  expression  for  the  central  angle 
<f>,  we  shall  proceed  as  follows:  The  distance  from  the  vertex  A  of  the 
spherical  surface  of  the  point  where  the  incident  ray  meets  the  yz- 
plane  of  co-ordinates  is  approximately  equal  to  V  p2  +  g2,  and  since 
the  length  of  the  arc  A  B  is  equal  to  r<p,  we  may,  if  we  neglect  the  mag- 
nitudes of  the  3rd  order,  put: 

rV  =  P*  +  22; 
and,  hence,  we  obtain: 


4- 


- 

r(J-J)*        u2  u2  uu 

We  must  now  derive  an  expression  for  cos2  p'  —  cos2  ju. 


462  Geometrical  Optics,  Chapter  XII.  [  §  319. 

The  approximate  equations  of  the  incident  ray  B  H  are  : 


n          b  c 

and  the  equations  of  the  straight  line  CH  are: 

y       z 

V     —      y  .       —     -?^     _ 

Jd     —     /  j  y 

y*    2* 

and  hence  for  the  angle  n  between  these  two  straight  lines,  we  have 


Now  since  the  point  H(r,  yh,  zh)  is  a  point  on  the  incident  ray,  we  have 


zh  =  —      a  =  -= 

n  J—J\     u          u)      J—J 

if,  for  the  sake  of  brevity,  we  write  temporarily  : 

Y-ji-jZ,    Z  =  J*-J* 

u         u  u         u 

Hence,  since  by  formula  (346): 


V-—Y     7'-—  7 
Y   ~  n'Y'     Z  ~  ri^ 


we  obtain: 

cos"  u  = 


,  ^  {(y  - y)Y  +  (z  -  z)Z}* 

(j-jfuV          y  +  z2 

/2  (ft  /\  Tr     i       /    l 


COS    /*'  = 

Now  evidently : 


and  hence  we  find : 

j 


sM'       sM=(./_y)2-pr 


§  320.]  Theory  of  Spherical  Aberrations.  463 

where  for  brevity  we  write: 


Now 


II          2  I         2/ 

-+—,  =  -  —/•A-  —  — , 

u      u'      r  n       n 

-  J_—  -2  —  7      I  __?Z, 

and  thus  we  can  write: 


n        n  u  r  n       n 


-2^         -     -        -  -JJ-A-  --         K.       (349) 
wu      \      r  n         n   /J 

Accordingly,  we  obtain  finally: 

cos2/  -  cos2  n  =  jj |r-a  •  A  -  -4,  (350) 

\j  —  j)       n 

where  A  is  defined  by  (349). 

320.     If  now  we  substitute  in  formula  (347)  the  expressions  (348) 
and  (350),  we  shall  obtain  on  the  right-hand  side  of  the  equation: 


where 

y 

R  =  - 


»3_    (j*     ,y\  R 

-J)3\Ju~Ju)'R' 


n         n        n 


n         n       n 
which  latter  expression  may  also  be  written  as  follows: 


A 

~rA—  --- 
J     nu  J 


/)2i    i\ 

--  A-  J 
r     n  J 


u  nu  uu  nu 

Thus,  we  obtain  finally  : 


.  (  0 


464 


Geometrical  Optics,  Chapter  XII. 


[§321. 


and,  similarly: 


n-5z\ 
-^)  = 


(353) 


where  R  is  defined  by  (351). 

321.  Thus  far  the  directions  of  the  axes  of  y  and  z  are  entirely 
arbitrary,  except  that  it  has  been  assumed  they  are  both  perpendicular 
to  the  optical  axis.  We  may  select  as  the  xy  -plane  the  meridian  plane 
which  contains  the  point  Q,  and  which,  according  to  GAUSS'S  Theory, 
will  contain  also  the  conjugate  point  Q'.  This  evidently  will  not 
affect  at  all  the  generality  of  the  treatment,  and  it  will  lead  to  some 
simplification,  inasmuch  as  we  shall  have  then  z  =  z'  =  o.  Thus 
if  we  put  2  =  o  in  the  formulae  (351),  (352)  and  (353),  we  obtain  the 
following  set  of  formulae  : 


y*(J2      i        (J-J)2i      i\ 

±-2[  —  A— -—  -A-  ) 

u  \  J     nu  J        r     n  / 


u 


nu 


yy 

- 

uu 


nu 


The  y-abenation: 


The  z-aberration: 
n-dz" 


(354) 


These  formulae  give  the  variations  of  n-8y/u,  n-dz/u  which  result  in 
consequence  of  the  refraction  of  the  ray  at  a  single  spherical  surface. 
In  case  we  have  a  centered  system  of  m  spherical  surfaces,  we  must 
introduce  the  subscript  k  to  indicate  that  the  formulae  apply  to  the  kth 
surface,  and  then  the  formulae  will  be  written : 


where 


/3 


§  321.]  Theory  of  Spherical  Aberrations.  465 

Now  if  h,  h  denote  the  incidence-heights  of  a  pair  of  paraxial  rays 
emanating  from  the  axial  object-points  Mv  Mlt  respectively,  we  have, 
by  ROBERT  SMITH'S  Law  (§  194): 


Moreover,  we  have  also  SEIDEL'S  Formula  (Chapter  VIII,  §195) 


If  we  introduce  these  relations  in  the  above  equations,  we  shall 
obtain  the  following  formulae : 

fry 

n—  I  =- 


*  (355) 


31 


Geometrical  Optics,  Chapter  XII. 


[§321. 


466 
Now 

or 


and,  hence,  if  we  suppose  that  the  object,  situated  in  the  first  medium, 
is  free  from  aberration,  so  that  the  object-point  Px  coincides  with  Qlt 
and  therefore 

we  find: 


*4.A(^y)  =  hkn^  -  h-S^f^; 


that  is, 


and,  similarly: 


Let  us  now  employ  the  following  abbreviations: 

Ar=n»    T.4  /     T     \ 

=  ^  TT  ^&*^  (  :  :  )  » 
*=i  hi  \nu  Jk 


2     r  2 


k=m    1.2 

civ  _  v  .13.    '  <  r 

"  7  " 


_ 


^^Alf./,       ..J),LA('LNj_^A/_L\  I: 

fe  *,  ssl  /,  (7*   /4)  r/U  A  /.  AU«  A  /  . 


(356) 


§  322.]  Theory  of  Spherical  Aberrations.  467 

so  that  we  may  write  finally  : 


i  3  L  ci  11     2     t  clr 

Q  Ut  7     o    —  ,  TO  U,U,  ;   'o 

3      l  l 


/         —         /  \Q      t  7  ,  O     ,, 

Um  2    (l*!-^)3      lhm  (Ut-Utf          lh 


322.  Conditions  of  the  Abolition  of  the  Spherical  Aberations  of 
the  3rd  Order.  The  expressions  denoted  here  by  S1,  Sn,  Sm,  5IV, 
5V  are  practically  equivalent  to  the  famous  five  sums  of  SEIDEL, 
although  SEIDEL'S  expressions  in  their  final  form  are  different  from 
these. 

The  equation  S1  =  o  will  be  recognized  as  the  condition  of  the 
abolition  of  the  spherical  aberration  at  the  centre  of  the  visual  field; 
that  is,  the  condition  that  the  axial  points  Mlt  M'm  shall  be  a  pair  of 
"aberrationless"  points  (§  265). 

The  equation  Su  =  o  is  at  the  same  time  the  condition  of  the  fulfil- 
ment of  ABBE'S  Sine-Condition  (§  284)  and  of  the  abolition  of  Coma 

(§313). 

The  condition  of  the  abolition  of  the  astigmatism  of  narrow  oblique 
bundles  of  rays  is  Sm  =  o  (§  302),  and  the  conditions  necessary  for 
a  plane,  stigmatic  image  are  Sm  =  o  and  5IV  =  o;  see  formulae  (332), 
§304. 

Finally,  the  condition  that  the  image  shall  be  without  Distortion 
is  5Y  =  o;  see  formula  (311)  or  formula  (312),  §  292. 

The  image  will  be  perfectly  faultless  (except  for  residual  errors  of 
the  5th  order)  provided  all  five  sums  S1,  S11,  5ln,  5IV,  and  Sv 
vanish  together,  and  these  five  conditions  are  necessary  if  the  image 
is  to  have  this  degree  of  perfection  in  every  respect. 

-SEIDEL'S  Formulae  (357),  which  give  the  magnitudes  of  the  y-  and 
2-aberrations  of  the  3rd  order  in  the  image-plane  <r'm,  are  derived  by 
A.  KERBERI  by  the  employment  of  KERBER'S  Formulae  given  in  Chap- 
ter IX,  §§  214,  216  for  the  refraction  of  a  ray  at  a  spherical  surface; 
wherein  the  trigonometrical  functions  are  replaced  by  their  series- 

1  A.  KERBER:   Beitraege  zur  Dioptrik.     Zweites  Heft  (Leipzig,  1896);  pages  9-15. 


468  Geometrical  Optics,  Chapter  XII.  [  §  323. 

developments.     KERBER'S  process  is  also  given  by  KOENIG  and  VON 
RoHR1  in  their  treatise  on  the  Theory  of  Spherical  Aberrations. 

ART.  103.    ELIMINATION  OF  THE  MAGNITUDES  DENOTED  BY  h,  u. 

323.  The  natural  determination-data  of  an  optical  system  are  the 
radii  (r)  of  the  spherical  surfaces,  the  thicknesses  (d)  of  the  intervening 
media  and  the  refractive  indices  (n).  If  in  addition  to  these  magni- 
tudes we  know  also  the  positions  of  the  object  and  of  the  stop,  which 
is  equivalent  to  knowing  the  values  of  u±  and  ult  we  can  compute  the 
values  of  the  two  systems  of  magnitudes  h,  u  and  h,  u  which  occur  in 
SEIDEL'S  Aberration-Formulae  (357).  So  long  as  these  formulae  are 
to  be  employed  to  investigate  the  defects  of  an  image  produced  by  a 
given  optical  system,  they  answer  their  purpose  excellently.  But  in 
case  the  problem  is  to  design  an  optical  instrument  which  is  to  fulfil 
certain  prescribed  conditions,  the  fact  that  the  equations  contain  two 
sets  of  magnitudes  which  are  not  independent  of  each  other  is  a  dis- 
advantage which  must  be  got  rid  of  by  eliminating  one  of  these  sets 
of  magnitudes  by  means  of  the  other  set.  In  SEIDEL'S  final  forms  of 
the  aberration-formulae  the  magnitudes  denoted  here  by  h,  u  do  not 
appear. 

This  elimination  is  performed  with  the  aid  of  the  two  formulae  (155) 
and  (156)  of  Chapter  VIII,  which  are  also  due  to  SEIDEL,  and  which, 
by  the  introduction  of  the  convenient  abbreviating  symbol  T,  may 
be  written  here  as  follows: 


T  =  hjik(jk  -  Jk)  =  M«i(/.  -  J,), 
h"  *       ^-' 


~"      '  It       Z.          * 

=2nk-l  •»&•»*-! 


(358) 


The  magnitude  denoted  here  by  T  depends  only  on  the  initial  values 
of  the  magnitudes  h,  u  and  h,  u.  If  we  introduce,  also  by  way  of 
abbreviation,  another  symbol  and  write  : 


k-i 


(359) 


formulae  (358)  may  be  put  in  the   following  forms   convenient  for 
direct  application  to  the  expressions  contained  in   the   aberration- 

1  A.  KOENIG  und  M.  VON  ROHR:  Die  Theorie  der  sphaerischen  Aberrationen:  Chapter 
V  of  M.  VON  ROHR'S  Die  Theorie  der  optischen  Inslrumente,  Bd.  I  (Berlin,  1904),  pages 
317-323. 


§  323.]  Theory  of  Spherical  Aberrations.  469 

formulae  (357): 

"ft  SB  **  |   ^    V^  "•         It'     \   ft-  J     f  '  I 

(360) 

Proceeding  now  to  eliminate  the  magnitudes  hk,  uk  from  the  express- 
ions under  the  summation-signs  in  the  formulae  (356),  we  remark, 
in  the  first  place,  that  the  sum  Sl,  which  is  the  expression  of  the  Co- 
efficient of  the  Spherical  Aberration  along  the  axis,  does  not  contain 
these  magnitudes  at  all.  Passing,  therefore,  to  the  Coma-Co-efficient, 
we  obtain  from  the  second  of  equations  (360) : 

f,i 

and  hence: 

The  first  of  the  two  terms  on  the  right-hand  side  of  this  equation  is 
the  co-efficient  Sl  which  is  concerned  with  the  spherical  aberration 
along  the  axis.  If  the  optical  system  satisfies  ABBE'S  Sine-Condition, 
it  must  be  spherically  corrected  for  the  object-point  Ml  (§277 
and  §  279) ;  that  is,  S1  =  o;  consequently,  the  formula  for  ABBE'S  Sine- 
Condition,  which  is  identical  with  what  SEIDEL  has  called  the  FRAUN- 
HOFER- Condition  (§284),  is: 


Again,  we  find: 
hence,  for  the  Astigmatic-Co-efficient  : 


L2 

**l  7,4  T2/_      i      v'  ^ 

=  -     hkjk(i  +  Xk) 


(363) 

and,  since 

hlhl(jk  -  J,)1  =  r2, 

we  find  also: 

rl        }<=m  T    \ 

(364) 


470  Geometrical  Optics,  Chapter  XII.  [  §  324. 

The  co-efficients  of  the  expressions  for  the  curvatures  of  the  two 
image-surfaces  formed  by  the  infinitely  narrow  pencils  of  meridian  and 
sagittal  rays  can  be  obtained  by  combining  the  two  equations  (363) 
and  (364). 

Finally,  since 


and 


k       "I 
we  have  the  following  expression  for  the  Distortion-Co-efficient  : 


nu     •  (365) 

k=l  k  k          k=l  "1  nU/k 

ART.  104.     REMARKS  ON  SEIDEL'S  FORMULAE:    AND  REFERENCES  TO 
OTHER   GENERAL   METHODS. 

324.  In  a  masterly  discussion  of  his  formulae,  SEIDEL  draws  also  a 
number  of  important  conclusions  of  a  general  kind,  which,  however, 
can  only  be  referred  to  here  very  briefly.  Thus,  for  example,  he  points 
out  that  it  is  impossible  (except  in  certain  special  cases  that  have 
comparatively  little  practical  interest)  to  construct  an  optical  appara- 
tus which  will  produce  a  correct  image  of  the  3rd  order  for  all  distances 
of  the  object.  If  it  is  required  to  form  such  images  of  objects  at  all 
distances,  in  addition  to  SEIDEL'S  five  equations  we  shall  have  other 
conditions  also,  one  of  which,  known  as  HERSCHEL'S  Equation,  is,  in 
general,  in  curious  contradiction  to  the  so-called  FRAUNHOFER-  or 
Sine-Condition  expressed  by  formula  (362)  :  so  that  the  two  conditions 
can  be  satisfied  at  the  same  time  only  in  particular  cases,  one  of  which 
is  that  the  image  shall  be  of  the  same  size  as  the  object. 

An  image  of  this  degree  of  perfection  even  in  the  case  of  one  special 
object-distance  can  only  be  attained  by  combining  in  the  system  of 
lenses  a  sufficient  number  of  separated  surfaces.  If  the  distances  be- 
tween the  spherical  surfaces  are  all  so  small  as  to  be  negligible  (so 
that  in  the  formulas  we  may  put  dk  =  o),  it  is  easy  to  show  that  the 
conditions  of  the  abolitions  of  all  the  errors  of  the  3rd  order  are  as 
follows  :  « 

S72'A  —  =  o,  (abolition  of  aberration  along  axis); 
2/»  A  —  =  o,  (abolition  of  comatic  aberration)  ; 


§  326.]  Theory  of  Spherical  Aberrations.  471 

n mum  =  n\u\y     ^^^n  =  °»  (condition  of  plane,  stigmatic  image) ; 
n'm  —  n\  =  o,  (abolition  of  distortion). 

This  last  condition  is  compatible  with  the  condition  nmum  =  n^  only 
in  case  the  optical  system  is  a  plane  mirror  or  an  infinitely  thin  plate 
of  glass:  and,  hence,  for  an  optical  system  which  shall  produce  images 
of  the  3rd  order  it  is  necessary  that  some  of  the  d's  at  least  shall  be 
different  from  zero. 

325.  In  connection  with  the  excellent  exposition  of  SEIDEL'S  theo- 
ries which    is    given  by  Professor  SILVANUS  P.  THOMPSON    in    an 
appendix  to  his  English  Translation  of  Dr.  O.  LUMMER'S  Beitraege  zur 
photo graphischen  Optik,1  he  directs  attention  to  a  remarkable  memoir 
published  by  FINSTERW ALDER2  in  1892,  wherein  the  author,  employing 
SEIDEL'S  Formulae,  derives  the  equation  of  the  Focal  Surface,  which 
is  the  envelope  of  the  bundle  of  emergent  rays  which  have  their  origin 
at  a  point  outside  the  optical  axis  of  a  centered  .system  of  spherical 
surfaces,  and    proceeds  then   to  show  in  a  very  simple  and  elegant 
manner  how  the  definition  of  the  image  and  the  distribution  of  the 
light  in  it  depends  on  the  extent  of  the  visual  field  and  on  the  aperture 
of  the  system  and  also,  in  the  case  when  the  image  is  real,  on  the 
position  of  the  focussing  screen.3     FINSTERWALDER  not  only  obtains 
by  his  method  results  which  are  in  complete  accord  with  those  of 
SEIDEL,  but,  as  Professor  THOMPSON  states,  he  has  "also  investigated 
the  distribution  of  the  light  in  the  coma,  and  its  changes  of  shape 
when  the  position  and  size  of  the  stop  are  changed". 

326.  With  regard  to  other  general  methods  of  investigation  in 
Optics,  the  following  paragraphs,  also  quoted  from  Professor  THOMP- 
SON'S chapter  on  "SEIDEL'S  Theory  of  the  Five  Aberrations",  may  be 
appropriately  inserted  at  this  place : 

1  O.  LUMMER:  Beitraege  zur  photographischen  Optik:  Zft.  f.  Instr.,  xvii  (1897),  208- 
219;  225-239;  264-271. 

SILVANUS  P.  THOMPSON:  Translation  of  OTTO  LUMMER'S  Contributions  to  Photographic 
Optics   (London,    1900). 

2  S.  FINSTERWALDER:  Die  von  optischen  Systemen  groesserer  Oeffnung  und  groesseren 
Gesichtsfeldes  erzeugten  Bilder:  Muench.  Abhand.  der  k.  bayer.  Akademie  der  Wiss.  II 
Cl.,  XVII  Bd.,  Ill  Abth.,  519-587.     Published  also  separately  in  Muenchen  in  1891  by 
G*  FRANZ. 

3  SEIDEL  himself  had  already  determined  the  equation  of  the  Focal  Surface,  without, 
however,  showing  how  the  equation  was  obtained.     See  SEIDEL'S  paper  entitled:  Ueber 
die  Theorie  der  caustischen  Flaechen,  welche  in  Folge  der  Spiegelung  oder  Brechung  von 
Strahlenbuescheln  an  den  Flaechen  eines  optischen  Apparates  erzeugt  werden:  Gelehrte 
Anzeigen  k.  bayr.  Akad.  d.  Wiss.,  xliv  (1857),  241-251.      See  also  a  letter  written  by 
SEIDEL  to  KUMMER,  and  published,  so  FINSTERWALDER  states,  in  Sitzungber.  der  k.  Akad. 
d.  Wiss.  zu  Berlin,  1867. 


472  Geometrical  Optics,  Chapter  XII.  [  §  326. 

''Remarkable  as  these  researches  of  VON  SEIDEL  are,  it  is  of  interest 
to  note  that  an  even  more  general  method  of  investigation  into  lens 
aberrations  had  been  previously  propounded.  This  is  the  fragment- 
ary paper  of  Sir  W.  ROWAN  HAMILTON,1  introducing  into  optics 
the  idea  of  a  'characteristic  function*  [see  §39],  namely  the  time  taken 
by  the  light  to  pass  from  one  point  to  another  of  its  path.  True,  he  did 
not  work  out  the  relations  between  the  constants  of  his  formulae  and 
the  data  of  the  optical  system.  Yet  the  method,  as  a  mathematical 
method  of  investigation,  is  unquestionably  more  powerful.  It  has 
recently,  and  independently,  been  revived  by  TniESEN,2  whose  equa- 
tions include  those  of  VON  SEIDEL. 

"The  latest  development  of  advanced  geometrical  optics  is  due  to 
Professor  H.  BRUNS,3  who  has  shown  that  in  general  the  formulas 
that  govern  the  formation  of  images  can  be  deduced  from  an  originating 
function  of  the  co-ordinates  of  the  rays — a  function  termed  by  him 
the  eikonal — by  differentiating  the  same,  just  as  in  theoretical  mech- 
anics the  components  of  the  forces  can  be  deduced  by  differentiation 
from  the  potential  function.  BRUNS'S  work  is  based  upon  the  theory 
of  contact-transformations  of  SOPHUS  LIE.  But  as  yet  neither  the 
formulae  of  BRUNS  nor  those  of  THIESEN  have  been  reduced  to  such 
shape  as  to  be  available  for  service  in  the  numerical  computation  of 
optical  systems." 

In  this  connection  it  may  be  stated  that  the  applications  of  SEIDEL'S 
aberration-formulae  to  the  calculation  and  design  of  optical  systems 
are  attended  with  much  difficulty,  and  on  this  account  practical 
opticians  seem  still  to  prefer  to  resort  to  the  methods  of  trigono- 
metrical calculations  of  the  paths  of  the  rays,  whereby  with  relatively 
less  trouble  they  arrive  at  safer  results  and  are  also  able  to  keep  track 
more  easily  of  the  effects  of  each  single  surface.  The  complete  solu- 
tion of  the  SEIDEL  formulae  is  indeed  only  possible  in  the  case  of  sys- 
tems of  comparatively  simple  structure.  The  greatest  practical  value 
of  these  general  formulae  is  to  guide  the  optician  to  a  correct  basis 
for  the  design  of  his  instrument  and  to  supply  him,  so  to  speak,  with 
a  starting-point  for  a  trigonometrical  calculation  of  the  particular 

lOn  some  Results  of  the  View  of  a  Characteristic  Function  in  Optics,  B.  A.  Report 
for  1833,  p.  360. 

1  M.  THIESEN:  Beitraege  zur  Dioptrik:  Berl.  Ber.,  1890;  799-813.  See  also:  Ueber 
vollkommene  Diopter:  WIED.  Ann.  (2)  xlv  (1892),  821-823;  Ueber  die  Construction 
von  Dioptern  mit  gegebenen  Eigenschaf ten :  WIED.  Ann.  (2)  xlv  (1892),  823-824. 
Also,  J.  CLASSEN:  Mathematische  Optik  (ScHUBERTsche  Sammlung  40),  Leipzig,  1901, 
Chapter  XI  entitled  "THIESENS  Theorie  der  Abbildungsfehler." 

3  H.  BRUNS:  Das  Eikonal:  Abhandlungen  der  math.-phys.  Cl.  der  k.  saechsischen  Akad. 
d.  Wiss.,  xxi  (1895),  321-436.  Also  published  by  S.  HIRZEL,  Leipzig,  1895. 


§  326.]  Theory  of  Spherical  Aberrations,  v  473 

system  which  he  aims  to  achieve.  Concerning  the  use  of  these  formulae 
the  reader  is  referred  to  a  valuable  and  interesting  article  by  A. 
KOENIG,  entitled  Die  Berechnung  optischer  Systeme  auf  Grund  der 
Theorie  der  Aberrationen.1 

In  a  series  of  learned  papers  C.  V.  L.  CHARLIER2  has  given  also  a 
method  of  investigating  the  spherical  aberrations  of  a  centered  system 
of  spherical  surfaces,  which  is  said  to  be  especially  adapted  to  the 
practical  design  of  optical  instruments.  But  it  is  impossible  here  to 
do  more  than  merely  refer  to  this  work. 

'See  Chapter  VII  (pages  373-408)  of  Die  Theorie  der  optischen  Instrumente,  Bd.-I 
(Berlin,  1904),  edited  by  M.  VON  ROHR.  See  also  A.  KERBER'S  Beitraege  zur  Dioptrik, 
published  in  Leipzig  from  1895  to  1899. 

2  C.  V.  L.  CHARLIER:  Ueber  den  Gang  des  Lichtes  durch  ein  System  von  sphaerischen 
Linsen:  Upsala,  Nova  Acta,  xvi  (1893),  1-20;  Zur  Theorie  der  optischen  Aberrations- 
curven:  Astr.  Nachr.,  cxxxvii  (1895),  No.  3265,  1-6;  Entwurf  einer  analytischen  Theorie 
zur  Construction  von  astronomischen  u.  photographischen  Objectiven:  Vierteljahrsschrift 
der  astronomischen  Gesellschaft,  31.  Jahrgang  (1896),  Leipzig,  pages  266-278.  See  also  a 
paper  by  R.  STEINHEIL:  Ueber  die  Berechnung  zweilinsiger  Objektive:  Zft.  f.  Instr.,  xvii 
(1897),  338-344,  in  which  the  writer  says  that  "  Die  Arbeit  des  Hrn.  CHARLIER  bedeute 
einen  Schritt  vorwaerts." 


CHAPTER    XIII. 

COLOUR-PHENOMENA. 
I.     DISPERSION  AND  PRISM-SPECTRA. 
ART.  105.     INTRODUCTORY  AND  HISTORICAL. 

327.  Relation  between  the  Refractive  Index  and  the  Wave- 
Length.  In  the  preceding  chapters  it  has  been  tacitly  assumed  that 
the  index  of  refraction  (n)  of  an  isotropic  optical  medium  was  a  con- 
stant magnitude;  which  assumption  was  permissible  so  long  as  we 
were  concerned  only  with  light  of  some  definite  kind  or  colour.  The 
length  (X)  of  a  light-wave  depends  on  two  factors,  the  speed  of  propa- 
gation (v)  and  the  vibration-number  or  frequency  (A/),  according  to 
the  familiar  formula: 

X  =  vfN. 

Light  of  a  definite  colour  is  characterized  by  a  definite  value  of  the 
frequency  N,  which  is  not  altered  when  the  light  is  refracted  from 
one  medium  into  another.  On  the  other  hand,  the  speed  (v)  with 
which  the  light  is  propagated  is  different  in  different  media,  and, 
consequently,  the  wave-length  (X)  must  vary  also.  However,  if  we 
select  some  standard  medium  (§  24),  as,  for  example,  the  free  ether  of 
empty  space  (wherein  also  light  of  all  colours  is  propagated  with  the 
same  speed),  the  wave-length  of  the  light  in  this  medium  may  be 
employed  also  to  characterize  the  colour  of  the  light.  In  this  chapter, 
therefore,  the  symbol  X  will  be  used  to  denote  always  the  wave-length 
of  the  light  in  vacuo. 

The  refractive  index  of  a  given  medium  is  a  function  of  the  wave- 
length X;  so  that  we  may  write: 


The  exact  character  of  this  relation  has  never  been  definitely  ascer- 
tained, although  a  number  of  formulae  have  been  proposed.  The 
earliest  and  best  known  of  such  formulae  is  the  one  suggested  by 
CAUCHY,1  as  follows: 

B       C 


where  A,  B,  C,  etc.,  denote  constants  depending  on  the  nature  of  the 
medium  and  diminishing  rapidly  in  magnitude  as  we  proceed  to  the 

1  A.  L.  CAUCHY:  Mimoire  sur  la  dispersion  de  la  lumiere;  published  in  Prague  in  1836. 

474 


§  328.]  Colour-Phenomena.  475 

higher  terms  of  the  series.  The  formula  shows  that  the  waves  of  the 
shorter  wave-lengths  are  the  more  highly  refracted.  In  media  which 
exhibit  the  so-called  phenomenon  of  "anomalous  dispersion"  it  is,  how- 
ever, not  true  that  the  shorter  waves  have  the  higher  indices  of  re- 
fraction, so  that  the  formula  is  by  no  means  general;  but  within 
certain  limits  it  is  found  to  represent  fairly  well  the  results  of  experi- 
ments. An  investigation  of  the  experimental  data  in  regard  to  this 
matter  shows  that,  in  general,  as  many  as  three  coefficients  A,  B,  C 
will  be  required  in  order  to  express  completely  the  relation  between 
n  and  X  for  all  optical  media;  although,  as  SCHMIDTI  has  shown,  in 
the  case  of  a  number  of  substances,  the  relation  may  be  right  well 
expressed  by  a  series  with  only  two  constants. 

We  see,  therefore,  that  until  we  specify  the  kind  of  light  that  is 
being  used,  the  refractive  index  of  a  medium  is  a  phrase  without  mean- 
ing; for  a  medium  has  just  as  many  indices  of  refraction  as  there  are 
different  kinds  of  light.  If,  for  example,  a  given  straight  line  is  the 
common  path  of  rays  of  two  or  more  kinds  of  light,  these  rays  will, 
in  general,  be  separated  by  refraction  and  made  to  take  different 
routes  when  they  enter  a  new  medium.  This  phenomenon  is  called 
Dispersion  of  the  Light,  sometimes  called  also  the  "chromatic  dis- 
persion". 

328.  Newton's  Prism-Experiments  and  the  Fraunhof  er  Lines  of  the 
Solar  Spectrum.  The  discovery  and  explanation  of  the  fact  that  the 
light  of  the  sun  is  composite  and  consists  of  light  of  a  great  variety  of 
colours  is  unquestionably  the  greatest  of  NEWTON'S  contributions  to 
optical  science.  Admitting  the  rays  of  the  sun  through  a  small  circular 
opening  in  the  window-shutter,  NEWTON  caused  these  rays  to  pass 
through  a  glass  prism,  and  was  surprised  to  find  that  the  image  on  the 
opposite  wall,  instead  of  being  a  circular  spot  of  white  light  (as  was 
produced  before  the  interposition  of  the  prism  in  the  path  of  the  beam) 
was  an  elongated  spectrum,  with  vivid  colours,  and  about  five  times 
as  long  as  it  was  broad.  NEWTON'S  remarkable  series  of  prism- 
experiments  was  begun  in  the  year  1666:  a  complete  description  .of 
them  was  afterwards  published  in  his  treatise  on  Optics.2  He  was 
led  to  conclude  that  sun-light  is  not  homogeneous,  but  is  composed 
of  rays  of  different  colours,  some  of  which  are  more  refrangible  than 
others,  the  red  rays  being  the  least  refracted  and  the  violet  rays  the 
most  refracted ;  so  that  the  coloured  spectrum  varied  by  impercept- 
ible gradations  of  colour  from  red  at  one  end  to  violet  at  the  other ; 

1  W.  SCHMIDT:  Die  Brechung  des  Lichts  in  Glaesern  (Leipzig,  1874). 

2  ISAAC  NEWTON:  Opticks:  or  a  treatise  of  the  reflexions,  refractions,  inflexions  and 
colours  of  light  (London,  1704).     The  discovery  of  Dispersion  and  the  explanation  of  the 
colours  of  the  Spectrum  was  communicated  to  the  Royal  Society  in  1672. 


476  Geometrical  Optics,  Chapter  XIII.  [  §  328. 

the  order  of  the  colours  (as  they  were  distinguished  by  NEWTON) 
being  red,  orange,  yellow,  green,  blue,  indigo  and  violet. 

The  important  practical  problem  of  abolishing,  if  possible,  the  chro- 
matic aberrations  of  optical  instruments,  especially  in  the  case  of  the 
telescope,  raised  the  question  as  to  whether  the  dispersions  of  dif- 
ferent substances  were  such  as  to  allow  of  combinations  which  neutral- 
ized the  dispersion  without  at  the  same  time  neutralizing  the  refraction. 
NEWTON  himself  conceived  that  he  had  proved  by  experiment  (Opticks, 
Book  i,  Part  ii,  Prop.  3)  that  achromatism  involved  necessarily  the 
abolition  of  ray-deviation  also;  so  that  in  an  achromatic  combination 
the  emergent  rays  must  needs  be  parallel  to  the  corresponding  inci- 
dent rays.  NEWTON  concluded,  therefore,  that  it  was  impossible  to 
produce  an  achromatic  image  by  refraction,  and  it  was  this  error 
that  "made  him  despair  of  improving  refracting  telescopes  and  led 
him  to  turn  his  attention  to  the  application  of  mirrors  to  these  instru- 
ments".1 NEWTON'S  authority  on  such  questions  was  so  great  that 
for  a  long  time  his  view  was  accepted  as  settling  the  matter. 

EuLER,2  approaching  the  subject  from  a  theoretical  stand-point, 
and  basing  his  argument  on  the  erroneous  assumption  that  the  human 
eye  is  an  achromatic  combination  of  lenses,  deduced  the  correct  con- 
clusion that  such  combinations  were  possible,  and  calculated  the  con- 
ditions that  were  necessary  therefor,  although  he  lacked  sufficient 
experimental  data.  In  1754  KLINGENSTIERNA,S  in  Sweden,  succeeded 
in  showing  by  a  combination  of  two  prisms  not  only  the  deviation  of 
the  rays  without  dispersion,  but  also  the  dispersion  of  the  rays  with- 
out deviation. 

HEATH4  states  that  the  mistake  in  NEWTON'S  experiment  (above 
referred  to)  "was  first  discovered  by  a  gentleman  of  Worcestershire 
named  HALL,  who  made  the  first  achromatic  telescope";  but  that 
"this  discovery  was  allowed  to  fall  into  oblivion,  until  the  experiment 
was  again  tried  by  DOLLOND,  an  optician  in  London,  who  found  that 
the  dispersion  could  be  corrected  without  destroying  the  refraction 
and  therefore  that  NEWTON'S  conclusion  was  not  correct".  In  1757, 
DOLLOND  was  able  to  construct  an  achromatic  telescope  by  the  use  of 
two  kinds  of  glass  called  "crown  glass"  and  "flint  glass",  of  which 
the  former  is  the  weaker  in  respect  to  both  refraction  and  dispersion. 

1  See  HEATH'S  Geometrical  Optics  (Cambridge,  1887),  Art.  179. 

2L.  EULER:  Sur  la  perfection  des  verres  obiectifs  des  lunettes:  Mem.  de  Berlin,  Hi 
(1747),  274-296. 

8  S.  KLINGENSTIERNA:  Anmerkung  ueber  das  Gesetz  der  Brechung  bei  Lichtstrahlen 
von  verschiedener  Art,  wenn  sie  durch  ein  durchsichtiges  Mittel  in  verschiedene  andere 
gehen:  Svensk.  Vel.  Acad.  Handl.,  xv  (1754),  300-306. 

*  HEATH'S  Geometrical  Optics  (Cambridge,  1887),  Art.  179. 


§  328.]  Colour-Phenomena.  477 

In  this  combination  the  convergent  lens  was  made  of  crown  glass  and 
the  divergent  lens  of  flint  glass. 

DOLLOND'S  success  revived  interest  in  the  question,  and  a  number 
of  mathematicians,  for  example,  EULER,  CLAIRAUT  and  D'ALEMBERT, 
proceeded  to  investigate  formulae  for  calculating  optical  systems;  but 
so  long  as  the  numerical  constants  of  the  different  kinds  of  glass  were 
not  available,  these  labours  were  necessarily  unproductive;  and  no 
farther  progress  worth  recording  was  achieved  until  the  era  of  FRAUN- 
HOFER  (1814),  whose  brilliant  researches  marked  the  dawn  of  a  new 
day  in  optical  science.  By  looking  through  a  prism  at  a  very  narrow 
slit,  formed  by  the  window-shutters  of  a  darkened  room,  WoLLASTON1 
had  detected  in  1802  that  the  solar  spectrum  was  crossed  by  dark 
bands  ;  but  it  was  not  until  these  so-called  FRAUNHOFER  Lines  were 
independently  re-discovered  by  FRAUNHOFER2  in  a  far  more  thorough 
and  scientific  manner  that  their  real  significance  and  value  were  recog- 
nized. 

In  the  Prism-Spectroscope,  such  as  was  afterwards  used  by  KIRCH- 
HOFF  and  BUNSEN,  the  source  of  the  light  is  an  illuminated  slit  placed 
parallel  to  the  edge  of  the  prism  in  the  focal  plane  of  a  collimating 
lens;  whereby  the  rays  incident  on  the  first  face  of  the  prism  are 
rendered  parallel.  If,  after  emerging  from  the  prism,  the  rays  are 
made  to  pass  through  a  second  convergent  lens,  there  will  be  formed  in 
the  focal  plane  of  this  lens  a  series  of  images  of  the  slit,  each  image 
corresponding  to  light  of  a  definite  colour  or  wave-length  (§  327). 
If  the  slit  is  illuminated  by  monochromatic  light,  there  will  be  only 
one  image,  but  if  the  incident  rays  are  composed  of  light,  say,  of  two 
kinds,  of  wave-lengths  Xj  and  X2,  we  shall  have  two  slit-images  side 
by  side  and  more  or  less  separated  from  each  other  depending,  among 
other  things,  on  the  magnitude  of  the  interval  XL  —  X2.  If 


the  two  slit-images  will  be  immediately  adjacent  to  each  other,  and 
they  may  partly  overlap  and  blur  each  other.  If  the  slit  is  illumi- 
nated by  white  light  emitted  originally  by  an  incandescent  solid, 
for  example,  the  light  of  an  electric  arc,  there  will  be  formed  in  the 

1  W.  H.  WOLLASTON:  A    method  of  examining  refractive  and  dispersive  powers,  by 
prismatic  reflection:  Phil.  Trans.,  ii  (1802),  365-380. 

2  A  preliminary  report  of  FRAUNHOFER'S  work  was  communicated  to  the  academy  of 
sciences  in  Munich  in  the  years  1814  and  1815.     See  also:  JOSEPH  FRAUNHOFER:  Bes- 
timmung  des  Brechungs-  und  Farbenzerstreuungsvermoegens  verschiedener  Glassorten, 
in  Bezugauf  die  Vervollkommnung  achromatischer  Fernroehre:  GILBERTS  Ann.,  Ivi  (1817), 
264-313- 


478  Geometrical  Optics,  Chapter  XIII.  [  §  329. 

focal  plane  of  the  receiving  lens  a  continuous  spectrum,  consisting  of  an 
innumerable  series  of  coloured  images  of  the  slit  of  every  gradation 
of  shade  from  red  to  violet,  one  image  for  each  of  the  infinite  varieties 
of  the  light  that  is  emitted  by  the  source.  A  definite  wave-length  (X) 
is  associated  with  each  colour,  and  to  each  wave-length  there  corre- 
sponds also  a  definite  value  of  the  refractive  index  (n) ,  which  increases 
continuously  from  its  greatest  value  for  the  extreme  red  light  to  its 
least  value  for  the  extreme  violet  light. 

However,  the  solar  spectrum  obtained  when  the  slit  is  illuminated 
by  sun-light  is  not  continuous,  as  NEWTON  supposed,  but  is  crossed 
by  a  vast  number  of  dark  bands  parallel  to  the  slit,  corresponding, 
as  we  know  now,  to  those  radiations  which  are  absent  from  the  light 
that  comes  to  us  from  the  sun.  It  would  be  more  correct  to  say  that 
these  dark  places  indicate  a  relative  deficiency  of  intensity  of  certain 
definite  kinds  of  light  in  what  we  call  sun-light.  These  FRAUNHOFER 
Lines  are  irregularly  distributed  over  the  entire  extent  of  the  solar 
spectrum,  and  although  their  actual  positions  will  be  altered  if  we 
replace  the  prism  of  the  spectroscope  by  another  one  of  different 
material,  the  order  of  the  lines  and  of  the  coloured  intervals  between 
them  is  always  the  same,  so  that  any  line  can  be  readily  recognized. 
The  great  importance  of  these  lines  for  optical  science  consists,  as 
FRAUNHOFER  was  quick  to  perceive,  in  the  fact  that  each  line  corre- 
sponds to  a  definite  wave-length  of  light,  and  hence  we  can  employ 
them  in  the  determinations  of  the  refractive  indices  of  a  substance. 
The  more  conspicuous  of  the  lines  in  the  different  parts  of  the  spectrum 
were  designated  by  FRAUNHOFER  by  the  capital  letters  of  the  Latin 
alphabet  from  A  to  H-,  the  violet  end  of  the  spectrum,  as  nearly  as 
he  could  locate  it,  being  designated  by  the  letter  /.  The  indices  of 
refraction  of  a  given  substance  for  rays  of  light  of  wave-lengths  corre- 
sponding to  the  FRAUNHOFER  Lines  A,  B,  C,  •  •  •  are  usually  denoted 
by  the  symbols  nA,  ns,  HC, 

329.  The  Jena  Glass.  Now  that  it  was  possible  to  determine  accu- 
rately the  optical  properties  of  different  media,  the  great  obstacle  in  the 
way  of  perfecting  optical  instruments  so  as  to  fulfil  as  far  as  possible  the 
theoretical  requirements  was  found  to  be  the  lack  of  suitable  kinds  of 
glass.  This  deficiency,  which  FRAUNHOFER  and  others  had  tried  to 
supply  by  the  manufacture  of  new  kinds  of  optical  glass,  began  to  be 
realized  more  and  more  with  the  development  of  the  microscope  and  in 
the  construction  of  the  photographic  objective.  Finally,  in  1881,  Pro- 
fessor E.  ABBE,  who  has  been  rightly  called  the  "GALILEO  of  the 
Microscope",  undertook,  in  conjuction  with  Dr.  O.  SCHOTT,  a  sys- 


§  329.]  Colour-Phenomena.  479 

tematic  investigation  of  the  "optical  properties  of  all  known  substances 
which  undergo  vitreous  fusion  and  solidify  in  non-crystalline  trans- 
parent masses".1  The  success  of  these  ingenious  and  exhaustive  ex- 
periments, in  which  entirely  new  and  remarkable  compositions  of 
glass  were  obtained  by  using  a  far  greater  number  of  chemical  elements 
than  had  ever  been  essayed  before  and,  especially,  by  employing  in 
the  manufacture  both  boric  and  phosphoric  acids  as  well  as  the  usual 
silicic  acid,  was  almost  immediate  and  beyond  all  expectations,  and 
a  few  years  later  (1886)  the  "Glastechnisches  Laboratorium"  of 
Messrs.  SCHOTT  und  Gen.,  in  Jena,  was  established,  where  the  now 
world-famous  "Jena  Glass"  is  manufactured. 

The  important  practical  problem,  suggested  first  by  FRAUNHOFER,  of 
producing  pairs  of  crown  glass  and  flint  glass  such  that  the  dispersions 
of  the  different  parts  of  the  spectrum  should  be  as  nearly  as  possible 
equal  for  both  kinds  of  glass,  with  the  object  of  abolishing  or  diminish- 
ing the  so-called  secondary  spectrum  (Art.  112),  was  successfully  solved 
by  the  labours  of  ABBE  and  SCHOTT.  Another  problem  of  not  less 
importance  consisted  in  producing  a  large  variety  of  kinds  of  optical 
glass  of  graduated  properties,  so  that  in  the  design  of  an  optical  system 
the  optician  might  be  able  to  find  a  combination  more  or  less  exactly 
adapted  to  his  particular  requirements.  This  result  was  likewise 
achieved. 

The  optical  properties  of  the  different  varieties  of  glass  are  de- 
scribed in  the  Jena-Glass  Catalogue  with  reference  to  five  bright 
lines  of  the  spectrum  which  are  all  easily  obtained  by  artificial  sources 
of  light,  viz.:  The  red  potassium  line,  which  is  very  close  to  the 
FRAUNHOFER  Line  A,  and  which  may  be  designated,  therefore,  by 
A'\  the  yellow  sodium  line  which  coincides  with  the  FRAUNHOFER 
Line  D\  and,  finally,  the  bright  lines  of  the  spectrum  of  hydrogen,  the 
first  two  of  which  are  identical  with  the  FRAUNHOFER  Lines  C  and  F, 
while  the  third,  designated  by  G' ,  is  very  near  the  FRAUNHOFER  Line 
G.  The  wave-lengths  of  the  light  corresponding  to  these  lines  are  as 
follows : 

1  See  E.  ABBE  und  O.  SCHOTT:  Productionsverzeichniss  des  glastechnischen  Labora- 
toriums  von  SCHOTT  und  Geno?sen  in  Jena:  published  as  a  "  prospectus  "  in  July,  i886t 
and  re-printed  in  Gesammelte  Abhandlungen  von  ERNST  ABBE,  Bd.  II  (Jena,  1906),  194- 
201.  See  also:  E.  ABBE:  Ueber  neue  Mikroskope:  Sitc,.-Ber.  Jen.  Ges.  Med.  u.  Nativ.,  1886, 
107-128;  reprinted  in  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904),  450-472. 

Especially,  see  S.  CZAPSKI:  Mittheilungen  ueber  das  glastechnische  Laboratorium 
in  Jena  und  die  von  ihm  hergestellten  neuen  optischen  Glaeser:  Zft.  f.  Inst.,  vi  (1886), 
293-299  and  335-348.  See  also  the  very  complete  history  of  optical  glass-manufacture 
given  in  M.  VON  ROHR'S  valuable  and  learned  work,  Theorie  und  Geschichte  des  pho- 
tographischen  Objektivs  (Berlin,  1899),  325-341. 


480 


Geometrical  Optics,  Chapter  XIII. 


[§329. 


X^'=  767.7  MM, 
\C  =  656.3  /iju, 
*D  =  589-3  MM, 
\F  =  486.1  MM, 
\G'=  434.1  MM- 

By  the  aid  of  these  data,  the  dispersion  of  the  glass  for  any  interval 
of  the  spectrum  comprised  between  lines  of  known  wave-lengths  may 
be  obtained,  closely  enough  at  any  rate  for  practical  purposes,  by  the 
method  of  graphical  interpolation,  wherein  the  abscissae  denote  the 
reciprocals  of  the  wave-lengths. 

The  following  list,  selected  somewhat  arbitrarily  from  the  'Table 
of  Optical  Glasses  made  in  Jena",  given  in  CZAPSKI'S  paper  in  the 
Zeitschrift  fur  Instrumentenkunde  (vi,  338-9),  will  serve  to  give  an 
idea  not  only  of  the  remarkable  range  and  variety  of  the  properties 
of  the  new  kinds  of  optical  glass,  but  also  of  the  fundamental  constants 
that  were  employed  by  ABBE  for  describing  these  properties: 
SOME  VARIETIES  OF  THE  JENA  OPTICAL  GLASS 


Factory 
Number 

Name 

nD 

Mean 
Dispersion 
nF~  nc 

ifj 

8      8 

II 

& 

Partial   Dispersions 

o  >> 

«c  -y 
'0  > 

&2 

c#G 

nD  ~~  nA' 

iip—  nD 

nG,  —nF 

0.225 

Light  Phos- 
phate-Crown 

I-SIS9 

0.00737 

70.0 

0.00485 
0.658 

0.00515 

0.698 

0.00407 
0.552 

2.58 

S.3o 

Heavy   Bari- 
um-Phosphate 
Crown 

1.5760 

0.00884 

65.2 

0.00570 
0.644 

O.OO622 
0.703 

O.OO5OO 
0.565 

3-35 

O.6o 

Calcium-Sili- 
cate-Crown 

I-5I79 

0.00860 

60.2 

0.00553 
0.643 

0.00605 
0.703 

0.00487 
0.566 

2.49 

0.138 

Sil.  Crown  of 
high  ref.  ind. 

1-5258 

0.00872 

60.2 

0.00560 
0.642 

0.00614 
0.704 

0.00494 
0.566 

2-53 

8.52 

Light  Borate- 
Crown 

1-5047 

0.00840 

6o.O 

0.00560 
0.667 

0.00587 
0.700 

0.00466 
0.555 

2.24 

S.35 

Borate-Flint 

1.5503 

0.00996 

55-2 

0.00654 
0.656 

0.00699 
0.702 

0.00561 
0.563 

2.56 

0.152 

Silicate  Glass 

I.5I59 

0.01049 

51.2 

0.00659 
0.628 

0.00743 
0.708 

O.OO6IO 
0.582 

2.76 

S.8 

Borate-Flint 

1.5736 

0.01129 

50.8 

0.00728 
0.645 

0.00795 
0.704 

0.00644 
0.571 

2.82 

0.164 

Boro-Silicate- 
Flint 

1.5503 

0.01114 

49.4 

O.OO7IO 
0.637 

0.00786 
0.706 

0.00644 
0.578 

2.8l 

S-7 

Borate-Flint 

1.6086 

0.01375 

44-3 

0.00864 
0.628 

0.00974 
0.708 

0.00802 
0.583 

3.17 

O.I54 

Light  Silicate- 
Flint 

I-57IO 

0.01327 

43-0 

0.00819 
0.617 

0.00943 
0.710 

O.OO79I 
0.596 

3.16 

S.57 

Heaviest  Sil.- 
Flint 

1.9626 

0.04882 

19.7 

0.02767 
0.567 

0.03547 
0.726 

0.03252 
0.666 

6.33 

§  329.]  Colour-Phenomena.  481 

The  index  of  refraction  of  each  kind  of  glass  for  the  D-Line  is  given 
in  the  first  column  of  the  table.  Since  this  line  is  about  at  the  bright- 
est part  of  the  spectrum,  and  since  also  this  radiation  is  especially 
convenient  to  obtain,  the  value  of  HD  is  usually  employed  to  charact- 
erize the  refrangibility  of  an  optical  medium. 

The  next  column  of  the  table  gives  the  value  of  the  so-called  mean 
dispersion,  that  is,  the  difference  (np  —  no)  of  the  indices  of  refraction 
for  the  light  corresponding  to  the  lines  C  and  F.  This  difference  is 
about  proportional  to  the  length  of  the  spectrum,  since  the  greater 
part  of  the  visible  spectrum  is  included  between  the  lines  C  and  F. 

The  third  column  gives  the  value  of  the  magnitude 


np  —  no 


(366) 


The  numerator  of  this  fraction  is  the  difference  between  the  mean 
index  of  refraction  (nj))  of  the  material  and  the  index  of  refraction  of 
air  (n  =  i);  which  difference  occurs  so  frequently,  for  example,  in  the 
formulae  of  Thin  Lenses.  The  reciprocal  of  this  fraction,  viz.,  i/v, 
is  called  the  relative  dispersion;  and,  hence,  the  greater  the  value  of  v, 
the  smaller  will  be  the  relative  dispersion.  It  will  be  remarked  that 
the  series  of  glasses  are  arranged  in  the  table  with  respect  to  the 
magnitude  of  this  constant  v  from  the  greatest  value  of  v  to  its  least 
value  in  descending  order.  This  is  due  to  the  fact  that  the  optical 
character  of  a  given  specimen  of  glass  is  seen  most  clearly  by  a  con- 
sideration of  its  p-value. 

The  values  of  the  partial  dispersions  for  the  three  intervals  A'-Dy 
D-F  and  F-G't  which  appear  in  the  next  three  columns  of  the 
table,  enable  us  to  perceive  also  the  behaviour  of  the  glass  as  regards 
dispersion;  so  that  we  can  compare  the  dispersions  of  two  different 
kinds  of  glass  for  the  various  parts  of  the  spectrum  with  a  view  to 
ascertaining  the  degree  of  achromatism  that  is  possible  by  a  combina- 
tion of  the  pair.  For  this  same  purpose  also  the  value  obtained  by 
dividing  the  partial  dispersion  of  one  of  these  intervals  by  the 
value  of  the  mean  dispersion  np  —  no  is  entered  in  the  same  col- 
umn immediately  under  the  value  of  the  partial  dispersion  to  which 
it  .belongs.  It  will  be  seen  from  the  table  that  the  partial  dispersions 
of  different  kinds  of  glass  are,  in  general,  quite  different.  Moreover, 
comparing  the  spectra  produced  by  two  different  optical  media,  we 
may  find  that  the  dispersion  of  the  red  region  is  relatively  greater, 
and  at  the  same  time  the  dispersion  of  the  blue  region  is  relatively 
less,  for  the  first  substance  than  the  corresponding  partial  dispersions 

32 


482  Geometrical  Optics,  Chapter  XIII.  [  §  329. 

for  the  second  substance.  This  phenomenon  is  known  as  the  irration- 
ality of  dispersion,  in  consequence  whereof  we  are  unable  to  compare 
the  spectra  produced  by  prisms  of  different  materials,  since  there  is 
no  law  of  proportionality  between  them.  This  fundamental  fact  in 
regard  to  prism-spectra  NEWTON  failed  to  perceive;  and  when  LUCAS 
of  Liege,  attempting  to  repeat  NEWTON'S  first  prism-experiment  (§  328), 
declared  that  he  could  never  obtain  a  spectrum  whose  length  was 
more  than  three  and  one-half  times  its  breadth,  NEWTON  persisted  in 
asserting  that,  if  the  experiment  were  properly  performed,  the  spec- 
trum would  be  found  to  be  five  times  as  long  as  it  was  broad;  whereas, 
no  doubt,  the  real  explanation  of  the  discrepancy  in  the  two  observa- 
tions was  to  be  found  in  the  fact  that  the  English  prism  and  the  Dutch 
prism  were  made  of  different  kinds  of  glass. 

By  comparing  the  corresponding  values  of  the  relative  partial  dis- 
persions of  two  different  specimens  of  glass,  say,  crown  and  flint,  we 
can  tell  immediately  what  will  be  the  character  and  extent  of  the 
residual  or  secondary  spectrum  obtained  by  a  combination  of  the  two 
materials.  Thus,  for  example,  a  large  value  of  the  relative  partial 
dispersion  for  the  interval  A'-D  will  mean  that  the  red  part  of  the 
spectrum  produced  with  this  kind  of  glass  will  be  relatively  extensive. 
The  difference  of  the  values  of  corresponding  ratios  for  two  specimens 
of  glass  will  be  a  measure  of  the  dissimilarity  of  the  two  spectra  in 
the  region  or  interval  to  which  the  ratio  applies.  On  the  other  hand, 
the  equality  of  these  corresponding  pairs  of  ratios  for  two  materials 
will  indicate  the  possibility  of  employing  these  kinds  of  glass  for  achro- 
matic combinations  that  are  free  from  secondary  colour-effects,  pro- 
vided also  the  ^-values  are  sufficiently  different  to  warrant  this  select- 
ion. Referring  to  the  table,  we  see  that  there  are  several  pairs  of 
varieties  of  the  Jena-glass,  which  have  approximately  equal  relative 
partial  dispersions  and  at  the  same  time  quite  different  rvalues,  and 
which,  therefore,  enable  us  to  make  achromatic  combinations  that  are 
practically  free  from  secondary  spectrum;  for  example,  the  pairs 
0.225  and  8.35;  8.40  and  8.35;  8.30  and  8.8;  and  O.6o  and  0.164. 
On  the  other  hand,  we  can  find  also  in  the  table  pairs  of  glasses  with 
approximately  equal  ^-values,  which  show,  however,  considerable  dif- 
ferences in  their  relative  partial  dispersions;  for  example,  compare 
0.138  and  8.52;  0.152  and  8.8;  and  8.7  and  0.154. 

Prior  to  the  time  of  ABBE,  there  was  no  kind  of  glass  available  for 
the  design  of  an  optical  instrument  which,  with  a  high  refractive  index, 
possessed  at  the  same  time  a  low  dispersive  power,  or  vice  versa.  Thus, 
for  example,  FRAUNHOFER'S  flint  glass  had  both  a  greater  refractive 


§  330.]  Colour-Phenomena.  483 

index  and  a  greater  dispersive  power  than  his  crown  glass.  But  a 
high  refractive  index  does  not  necessarily  imply  also  a  great  dispersive 
power,  as  was  formerly  supposed,  as  will  be  seen  by  comparing  the 
following  pair  of  products  of  the  Jena-Glass  Laboratory : 

nD  nF~nC 

0.1209.     Densest  Baryta  Crown  1.0112  0.01068 

0.7260.     Extra  Light  Flint  1-5398  0.01142 

Here  it  will  be  remarked  that  the  more  highly  refracting  of  these  two 
specimens  is  at  the  same  time  the  less  strongly  dispersive  one  of  the 
pair.  It  is  easy  to  understand  how  the  production  of  different  kinds 
of  glass  with  such  properties  as  we  have  noted  marked  an  epoch  in 
optical  engineering  and  made  possible  the  extraordinary  perfections 
of  modern  optical  instruments. 

330.     Combinations  of  Thin  Prisms. 

In  connection  with  this  subject  it  will  be  of  service  to  consider  here 
briefly  two  combinations  which  have  been  mentioned  above  and  which 
have  great  practical  importance,  viz.,  the  case  of  deviation  without  dis- 
persion and  the  case  of  dispersion  without  deviation.  Suppose  that  we 
have  two  prisms  made  of  substances  whose  indices  of  refraction  for 
light  of  a  given  wave-length  X  may  be  denoted  by  n  and  n'  \  and,  for 
the  sake  of  simplicity,  let  us  assume,  for  the  present,  that  the  refracting 
angles  ft  and  (3f  are  exceedingly  small,  and  also  that  the  rays  which  we 
employ  meet  the  surfaces  of  the  prisms  at  very  nearly  normal  incidence. 
Of  course,  these  assumptions  are  widely  different  from  the  conditions 
that  we  have  in  an  actual  case;  but  that  need  not  affect  the  object 
which  we  have  here  in  view. 

If  e  denotes  the  total  deviation  of  the  ray  of  wave-length  X  that  is 
produced  by  the  pair  of  prisms  in  combination,  then,  by  formula  (28) 
of  §  72,  we  can  write: 

€  =  (n  -  1)0  +  (n'  -  i)/3'.  (367) 

The  variation  de  of  the  deviation  in  consequence  of  a  variation  of 
the  wave-length  of  light  from  the  value  X  to  the  value  X  +  d\  will  be  a 
measure  of  the  dispersion.  Thus,  by  differentiation,  we  obtain: 

de  =  p-dn  +  p'-dn'.  (368) 

(i)  If  the  combination  of  the  two  thin  prisms  is  to  be  achromatic 
with  respect  to  light  of  wave-lengths  X  and  X  +  d\,  then  we  must 
put  de  =  o,  and,  hence,  the  condition  of  achromatism  requires  that 


484  Geometrical  Optics,  Chapter  XIII.  [  §  331. 

the  angles  0,  /3'  shall  be  related  as  follows: 

/?!-       ^L 
J  =    ~  dn'' 

In  order  to  obtain  with  this  combination  a  given  deviation 


of  the  D-ray,  we  must  have  therefore  : 

"-<„-•).*,•  f-  -(,-?).  A,"         (369) 

where 

i          ^n          1         dnr 

v-^~i'  v'=^=~i  (370) 

denote  the  so-called  relative  dispersions  of  the  two  optical  media. 
In  the  achromatic  prism-combination  it  is  usual  to  superpose  the  lines 
C  and  F  (red  and  blue)  ;  in  which  case: 

dn  =  nF  —  nG,     dn'  —  n'F  —  ric. 

(2)  On  the  other  hand,  if  we  are  to  have  dispersion  without  deviation 
(as  in  the  so-called  "direct-vision"  combination  of  prisms),  then, 
assuming  that  the  D-ray  is  the  ray  which  is  to  emerge  without  devia- 
tion (eD  =  o),  we  shall  have: 


_ 
ft          n'D  -  i 

and,  hence,  for  a  given  value  de  of  the  dispersion  of  the  rays  of  wave- 
lengths X  and  X  +  dX,  we  find  : 


where  v,  v'  are  the  magnitudes  defined  according  to  equations  (370). 

ART.  106.     THE    DISPERSION    OF   A    SYSTEM    OF    PRISMS. 

331.  When  a  ray  of  light  is  refracted  in  succession  through  a  series 
of  optical  media,  the  angular  deviation  e  is  a  function  of  the  indices  of 
refraction  nlt  n'lt  n'2,  etc.,  and  each  of  these  latter  magnitudes  is  itself 
a  function  of  the  wave-length  X.  The  change  of  the  angular  deviation 
corresponding  to  a  given  change  of  the  wave-length  X  is  a  measure  of 
the  dispersion  of  the  system  for  this  interval.  Accordingly,  the  dis- 


§  331.]  Colour-Phenomena.  485 

persion  for  the  interval  comprised  between  the  values  X  and  X  +  d\ 
will  be  expressed  analytically  by  the  following  formula: 

<3e        de   dn{        de    dn'2  de   dnm 

^'+7~'          +         ~' 


wherein  it  is  assumed  that  there  is  no  dispersion  of  the  light  in  the 
first  medium  (de/dnl  =  o).  In  this  formula  m  denotes  the  number  of 
refracting  surfaces.  The  partial  differential  co-efficients  dtjdn  are 
not  only  functions  of  the  refractive  indices  nlt  n[,  n'2,  etc.,  but  these 
magnitudes  depend  also  on  the  forms  and  position-relations  of  the 
refracting  surfaces;  whereas  the  magnitudes  dn/d\  depend  only  on 
the  form  of  the  function  connecting  the  variables  n  and  X  (§  327)  and 
on  the  values  of  the  numerical  constants  of  the  medium  in  question; 
and,  hence,  it  has  been  suggested  that  the  differential  co-efficient  dn/d\ 
might  properly  be  called  the  "characteristic  dispersion"  of  the  medium. 
Accordingly,  the  problem  of  finding  the  dispersion  in  the  case  of  a 
given  optical  system  consists  in  determining  the  values  of  the  magni- 
tudes de/dn  for  each  medium.  We  propose  now  to  investigate  this 
problem  in  the  case  of  a  system  of  prisms  with  their  refracting  edges 
all  parallel.1 

According  to  formulae  (43)  of  §93,  we  have,  for  the  refraction  at 
the  kth  plane  refracting  surface  of  a  ray  lying  in  a  principal  section 
of  the  prism-system,  the  following  equations: 


nk'smak  =  nk_l'smak,  1 

r 

€&  =  <*/<-<**;     J 


(373) 


where  nk  denotes  the  index  of  refraction  of  the  (k  +  i)th  medium  for 
light  of  the  given  wave-length  X  ;  a&>  ctk  denote  the  angles  of  incidence 
and  refraction  at  the  kth  surface  ;  and  ek  denotes  the  angular  deviation 
of  the  ray  produced  by  this  refraction.  Moreover,  if  f$k  denotes  the 
refracting  angle  of  the  kth  prism  (that  is,  the  dihedral  angle  between 
the  kth  and  the  (k  +  i)th  refracting  planes,  as  in  §  93),  we  have 
also: 

«*+i  =  «!-&;  (374) 

1  See  S.  CZAPSKI:  Theorie  der  optischen  Instrument  e  nach  ABBE  (Breslau,  1893),  pages 
145,  foil.;  H.  KAYSER:  Handbuch  der  Spectroscopie,  Bd.  I  (Leipzig,  1900),  Arts.  297, 
foil.;  and  F.  LOEWE'S  "  Die  Prismen  und  die  Prismensysteme  "  which  is  Chapter  VIII 
of  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR; 
pages  455-457- 


486  Geometrical  Optics,  Chapter  XIII.  [  §  331. 

whence  we  obtain  for  the  total  deviation  (e)  of  the  ray  of  wave-length  X: 

k=m  k=m-l 

k=l  &=l 

where,  as  above  stated,  m  denotes  the  total  number  of  plane  refracting 
surfaces. 

The  total  deviation  of  a  ray  of  wave-length  X  +  d\  will  be  €  -f-  de, 
where  according  to  the  formula  above: 

de  =  da^  —  dam. 

If,  as  is  usually  the  case,  there  is  no  dispersion  of  the  light  in  the  first 
medium,  that  is,  if  «L  has  the  same  value  for  the  rays  X  and  X  +  d\, 
then  we  must  put  dav  =  o,  in  which  case  we  have  therefore : 

de=  -  da'm. 

The  magnitude  de  is  a  measure  of  the  dispersion  of  the  light  of  wave- 
lengths X  and  X  +  d\. 

Differentiating  the  first  of  equations  (373),  we  obtain: 

nk  -  cos  ak  •  dak  +  sin  a'k  •  dn'k  =  nk_^  •  cos  ak  -  dak  +  sin  ak  -  dnk_lt 
wherein,  according  to  formula  (374),  we  have: 

dak  =  dak_l. 
-**. 

This  equation  may  evidently  be  written  in  the  following  recurrent 
form: 

,  _  nk-i    cosafe      ,  i 

/  .  nk     cos  ak     k~l      nk*cos  dk    kt 

where,  for  brevity,  we  have  written: 

Xk  =  nk_i -sin  «»(§*-  ^)  (375) 

\    fl>k  ilfi -i     /  . 

Thus,  we  obtain  the  following  formula: 


k-i 


wherein  it  is  to  be  understood  that  we  must  put  always  cos  am+l  =  i . 
In  this  equation  nk  and  nk  +  dn'k  denote  the  indices  of  refraction  of 
the  (k  +  i)th  medium  for  light  of  wave-lengths  X  and  X  +  dX, 
respectively. 


§  332.]  Colour-Phenomena.  487 

//  the  first  and  last  media  are  both  air,  we  can  put : 

»«  =  ni  =  I ! 
and,  if,  moreover,  there  is  no  initial  dispersion,  we  can  put  also: 

da^  =  O,     de  =  —  da'm. 
Accordingly,  under  these  circumstances,  we  have: 


(377) 


332.    Dispersion  of  a  Single  Prism  in  Air. 

Assuming  that  there  is  no  initial  dispersion  (dal  =  o)  and  that  the 
prism  is  surrounded  by  air,  so  that  we  may  write: 

ni  =  n2  —  I>     n{  =  n, 

and  putting  m  =  2  in  formulae  (377),  we  obtain  for  the  dispersion  of  a 
single  prism: 

£?!  sin  ft         <fa 

dX~cosa;-cosc4d\' 
where 


denotes  the  refracting  angle  of  the  prism.  According  to  this  formula, 
the  dispersion  of  a  single  prism  for  light  of  wave-lengths  X  and  X  +  d\ 
depends  not  only  on  the  value  of  the  refractive  index  n  but  also  on 
the  refracting  angle  jS  and  on  the  angle  of  incidence  a,.  When  the 
angle  of  emergence  a'2  =  90°,  the  dispersion  de/d\=oo  has  its  maximum 
value.  As  the  angle  a'2  decreases  (in  consequence  of  a  corresponding 
variation  of  the  incidence-angle  aj,  the  dispersion  de/d\  diminishes 
until  it  reaches  a  minimum  value,  after  which  farther  decrease  of  the 
angle  a2  is  accompanied  by  increase  of  the  dispersion.  The  fact  that 
for  a  certain  value  of  the  incidence-angle  o^  the  dispersion  de/d\  is  a 
minimum  was  first  remarked  by  J.  F.  W.  HERSCHEL/  who  found  also 
that  this  position  was  different  from  that  of  minimum  deviation. 
The  dispersion  will  be  a  minimum  for  that  value  of  the  incidence- 
angle  «!  for  which  cos  <x{  -  cos  a'2  is  a  maximum  ;  but  the  solution  of  this 

1  J.  F.  W.  HERSCHEL:  Article  "  On  Light  "  in  the  Encyc.  Metropolitana  (London, 

1828). 


488  Geometrical  Optics,  Chapter  XIII.  [  §  333. 

problem  leads  to  a  cubic  equation  for  the  determination  of  c^.1  Ac- 
cording to  THOLLON,2  the  condition  of  minimum  dispersion  is  given 
approximately  by  the  following  equation: 

«'.  =  -  «W  (379) 

If   the  prism  is  in  the  position  of  minimum  deviation  (e  =  e0),  we 
have  (see  §71): 


a    =  — 


and  if  these  values  are  introduced  in  formula  (378),  we  find  for  the 
dispersion  of  the  prism  in  this  special  position  : 

de0      2  dn 

''-  (38o) 


333.  Another  special  case,  which  is  of  interest  from  a  practical 
standpoint  is  the  dispersion  of  a  train  of  prisms  composed  alternately 
of  glass  and  air,  so  that  we  are  concerned  with  only  two  media.  Here 
also  we  shall  assume  that  there  is  no  initial  dispersion  (d^  —  o)  and 
also  that  the  dispersion  of  the  air-prisms  is  negligible.  If  we  put 

ni  —  nz  =  ' ' '  =  nzi  =  ' '  *  =  T>     nl  =  nz  =  •  •  •  =  w2»-i  =  •• '  =  w» 

we  shall  have  for  the  refraction  from  air  to  glass  at  the  (2i  —  i)th 
surface : 

X2i_ l  =  sin  cx2i_l'dnj 

and  for  the  next  following  refraction,  that  is,  from  glass  to  air: 

X2i  =  —  sin  a2i-dn, 

where  i  denotes  here  any  positive  integer  from  i  =  i  to  i  =  m/2. 
Substituting  these  values  in  formula  (377),  we  obtain: 

de       dn  (   .      ,  cos  a2  •  cos  «3  •  •  •  cos  ctm        .        cos  «3  •  cos  a4  •  •  •  cos  am 

•rr  =  "^r  \  sin  oil ' —     — ' ' —  sin  a.2 ' / 1 ? 

o\       a\  [  cos  al  •  cos  a2  •  •  •  cos  am  cos  a2  •  cos  o:3  •  •  •  cos  am 

,  cos  o?4  •  cos  QJS  •  •  •  cos  am 

+  sm  a, ; 1 T  —  •  •  •  \ . 

3  cos  <*3  •  cos  a4  •  •  •  cos  am  \ 

In  the  special  case  when  the  path  of  the  ray  through  the  system  of 
prisms  which  are  composed  alternately  of  glass  and  air  is  symmetrical, 

1  See  H.  KAYSER:   Handbuch  der  Speclroscopie,  Bd.  I  (Leipzig,  1900),  Art.  300. 
8  L.  THOLLON:  Minimum  de  dispersion  des  prismes;  achromatisme  de  deux  lentilles 
de  m^me  substance:   Comptes  Rendus,  Ixxxix  (1879),  93-97. 


§  334.]  Colour-Phenomena.  489 

we  shall  have: 


«1    =    -  <*2   =   «3   =    •  •  •    =   «2i-l   =    -  «2i   =•"    =    -  «,„; 

and  evidently  now  each  of  the  m  terms  within  the  brackets  on  the 
right-hand  side  of  the  above  equation  will  be  equal  to 

sin  a 


cos  at 

the  signs  of  the  terms  being  all  positive.  Since,  moreover,  w-sin  a{  = 
sin  «lf  we  obtain  for  the  magnitude  of  the  dispersion  under  these  cir 
cumstances  : 

de0      m  dn 


where  m/2  denotes  the  number  of  glass  prisms. 

Comparing  this  result  with  formula  (3  80) ,  we  see  that  the  dispersion 
of  a  train  of  glass  prisms  adjusted  as  above  described  is  equal  to  the 
sum  of  the  dispersions  of  the  prisms  taken  separately.  This  formula 
(381)  is  a  useful  one,  because  in  actual  practice  the  prisms  of  a  prism- 
spectroscope  are  usually  adjusted  in  this  way. 

334.     Achromatic  Prism-Systems. 

The  condition  that  the  rays  of  wave-lengths  X  and  X  +  d\  shall 
emerge  from  the  optical  system  along  the  same  identical  path  is 
de/d\  =  o;  in  which  case  the  deviations  of  the  two  rays  will  have  the 
same  value  e.  This  is  the  case  of  Deviation  Without  Dispersion. 
Assuming,  as  is  usually  the  case,  that  the  incident  rays  are  them- 
selves without  dispersion,  we  find  by  formula  (377)  the  following 
condition  of  achromatism  of  a  system  of  prisms  for  light  of  wave- 
lengths X  and  X  +  d\: 

£=?       y^coso;r+1  _      1 

*=1   *r=*  COSQV  "  (382) 

cosam+1  =  I.J 

If,  for  example,  the  system  is  composed  of  three  prisms  (m  =  4), 
and  if  the  first,  third  and  last  media  are  air  (^  =  n'2  =  n'4  =  i),  so 
that  the  system  consists,  let  us  say,  of  two  glass  prisms  separated  by 
air,  the  combination  will  be  achromatic  for  light  of  wave-lengths  X 
and  X  +  d\,  provided  we  have: 

sin  /31  •  cos  «3  •  cos  o?4  •  dn[  +  sin  /33  •  cos  a[  •  cos  <x'2  •  dn'3  =  o,      (383) 


490  Geometrical  Optics,  Chapter  XIII.  [  §  334. 

where 

0i  =  «i,—  «2»     ft  =  «2  —  «s 

denote  the  refracting  angles  of  the  two  glass  prisms.     In  this  equation 
the  magnitudes  a{j  a'2,  or3  and  «4  are  connected  by  the  relations: 


sn 


a2  =  n(  -sin  (a(  —  ft),     sin  a3  =  n'3  -sin  (<*4  +  ft)  ; 


and,  hence,  if  the  first  prism  is  supposed  to  be  known,  that  is,  if  the 
magnitudes  denoted  by  n(  and  ft  are  given,  and  if  also  the  angle  of 
incidence  aL  and  the  index  of  refraction  n'3  of  the  second  prism  are 
given,  there  will  still  remain  two  arbitrary  magnitudes,  viz.,  ft  and  <*3. 
Under  these  circumstances,  therefore,  the  condition  expressed  by  equa- 
tion (383)  may  be  satisfied  in  either  of  two  ways,  as  follows:  (i)  Any 
arbitrary  value  may  be  assigned  to  the  refracting  angle  (ft)  of  the 
second  glass  prism,  and  we  shall  have  then  to  determine  the  correspond- 
ing value  of  the  angle  «3,  that  is,  we  shall  have  to  find  the  orientation  of 
the  second  glass  prism  with  respect  to  the  first  in  order  that  the  com- 
bination may  be  achromatic;  or  (2)  Assuming  an  arbitrary  value  of 
the  angle  «3,  we  may  then  employ  equation  (383)  to  determine  what 
value  the  refracting  angle  of  the  second  glass  prism  must  have.  The 
two  glass  prisms  may  even  be  made  of  the  same  kind  of  glass  (n(  =  w3). 
As  a  concrete  illustration,  let  us  assume  that  the  ray  of  wave-length 
X  traverses  each  of  the  glass  prisms  symmetrically,  that  is,  with  mini- 
mum deviation;  in  which  case  we  have  the  following  relations  (§71): 

ft         >  ft 

«!  =  —  a2,     «!  =  -  «2  =  —  ,     a3  =  —  «4  =  —  . 

*•>  £ 

Introducing  these  values  in  equation  (383),  we  obtain  the  condition  of 
achromatism  for  this  special  case  in  the  following  form: 


where 


dn[  dn\ 

tan  a,  — -r  -f-  tan  a»  — —  =  o, 
3 


'0t  >          /         0i 

sm  a,  =  n,  -  sin  —  ,     sin  a3  =  Wo  •  sin  —  . 

2  3  3  2 


The  simplest  case  is  that  in  which  the  system  is  composed  of  two 
glass  prisms  (usually  cemented  together  along  their  common  face),  the 
first  and  last  media  being  air,  so  that  Wj  =  w3  =  I.  For  this  case 
m  =  3,  and  by  formula  (375)  we  find: 

Xv  —  sin  <*!  >dn(,     X2  =  sin  a'2'dn'2  —  sin  a^dn^     X3  =  —  sin  a^dn2-t 


§  335.] 


Colour-Phenomena. 


491 


whence,  employing  formulae  (377),  we  obtain  after  several  obvious 
reductions : 

sin  ft  •  cos  a3  •  dn(  -J-  sin  ft  •  dn'2 

.(384) 


de  = 


cos  a9  -  cos 

-  ** 

where  ft  =  a(  -  a2,     ft  =  a'2  -  <*3 

denote  the  refracting  angles  of  the  prisms. 

If,  therefore,  this  combination  is  to  be  achromatic  for  light  of  wave- 
lengths X  and  X  +  d\,  we  must  have : 

sin  ft -cos  a3'dn(  +  sin  ft- dn'2  =  o.  (385) 

By  means  of  this  formula,  the  angle  ft  of  the  second  prism  can  be 
calculated ,  so  soon  as  we  assign  the  value  of  the  angle  of  incidence  (a^ 
and  the  value  of  the  deviation-angle  (e). 

335.  Direct-Vision  Prism-System.  We  may  consider  briefly  also 
the  important  practical  case  of  a  system  which  is  constructed  so  that, 
although  the  rays  of  wave- 
lengths X  and  X  +  d\  are  dis- 
persed, the  standard  ray  of 
wave-length  X  traverses  the 
system  without  being  devi- 
ated (e  =  o) —  prism-system 
a  vision  directe.  If,  as  in  the 
special  case  considered  in 
§  334,  the  system  is  com- 
posed of  two  cemented  glass  FIG  149 
prisms  (m  =  3)  surrounded 
by  air,  the  dispersion  is  given 
by  formula  (384)  above.  If 
we  specialize  the  problem 
still  farther  by  supposing  that  the  ray  of  wave-length  X  emerges  from 
the  system  in  a  direction  perpendicular  to  the  third  plane  refracting 
surface  (Fig.  149),  we  have  evidently  the  following  system  of  equa- 
tions for  this  Direct-Vision  Combination: 

n,=n'=  i, 


(386) 


DIRECT-VISION  COMBINATION  OF  Two  CEMENTED 
GLASS  PRISMS.  The  portion  DEF  of  the  Crown-Glass 
Prism  can  be  cut  away,  as  no  rays  traverse  this  part. 
See  diagram  of  AMICI- Prism,  Fig.  153. 


sin  a2  =  -7  sin  ft, 
n\ 

ci  =  —  c2  =  «i  —  «I,     c3  =  c  =  O, 


tan  a,  =  — 


n  •  cos  c,  — 


492  Geometrical  Optics,  Chapter  XIII.  [  §  336. 

whence  the  magnitude  of  the  refracting  angle  /32  of  the  second  prism 
can  be  calculated. 

The  dispersion  of  the  system  is  given  by  the  following  formula: 

dn'L 


cos  fo  -  ft) 

When  a  beam  of  parallel  rays  is  refracted  at  a  plane  surface,  the 
widths  b,  b'  of  the  pencils  of  incident  and  refracted  rays  are  in  the 
same  ratio  to  each  other  as  the  cosines  of  the  angles  of  incidence  and 
refraction;  thus, 

bk        cos  ak 
b'k_i      cos  otk 

and,  hence,  also,  in  the  case  of  a  system  of  plane  refracting  surfaces: 

, 

(387) 


Applying  this  formula  to  the  prism-system  represented  in  Fig.  149, 
we  obtain: 

coscycosa., 


cos  (at!  -  €L)  -cos  (a2  +  ej* 


^ 


In  the  "direct-vision"  prism-system  designed  by  AMici,1  the  com- 
bination consists  of  three  prisms  cemented  together,  the  first  and 
third  of  which  are  precisely  alike  and  both  made  of  crown  glass; 
whereas  the  second,  or  middle,  prism  is  made  of  flint  glass.  If  the 
AMI  ci-sy  stem  is  divided  into  two  equal  halves  by  a  plane  containing 
the  refracting  edge  of  the  flint  glass  prism,  the  first  half  will  be  exactly 
similar  to  the  combination  represented  in  Fig.  149.  In  the  AMICI- 
combination  the  widths  of  the  incident  and  emergent  beams  of  rays 
are  equal. 

ART.   107.     PURITY   OF  THE   SPECTRUM.     RESOLVING   POWER   OF  PRISM- 

SYSTEM. 

336.  Measure  of  the  Purity  of  the  Spectrum.  In  the  investi- 
gation of  the  spectra  produced  by  a  prism-system,  the  problem 

1  J.  B.  AMICI:  Museo  florentino,  i  (1860),  p.  I.  (This  reference  has  not  been  verified 
by  the  author).  —  Concerning  AMICI'S  prism-system,  see  especially  S.  CZAPSKI:  Theorie 
der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  pages  149,  150  and  153;  and 
F.  LOEWE'S  "  Die  Prismen  und  die  Prismensysteme  ",  pages  461,  462  of  Die  Theorie  der 
optischen  Instrumente,  Bd.  I  (Berlin.  1904),  edited  by  M.  VON  ROHR.  Both  CZAPSKI  and 
LOEWE  give  a  formula  which  is  different  from  formula  (388)  above  and  which  the  author 
could  not  obtain. 


§  336.]  Colour-Phenomena.  493 


is  not  merely  to  increase  the  dispersion  de/dX,  but  rather  to  obtain 
as  nearly  as  possible  a  pure  spectrum,  wherein  the  light  to  be  ana- 
lyzed is  resolved  into  its  simplest  components,  so  that  at  any  given 
part  of  the  spectrum  the  difference  d\  of  the  wave-lengths  that  are 
superposed  shall  be  as  small  as  possible.  The  spectrum  is  composed 
of  a  series  of  images  of  the  slit,  each  of  which  corresponds  to  light 
of  a  definite  kind  or  colour;  and  if  the  apparent  width  of  the  slit- 
image  for  light  of  wave-length  X  is  greater  than  the  angular  dis- 
persion de  of  the  rays  of  wave-lengths  X  and  X  +  d\,  the  slit-images 
corresponding  to  these  two  radiations  will  partly  overlap  each  other, 
and,  accordingly,  the  spectrum  in  this  region  will  be  more  or  less 
impure.  If  the  slit  itself  were  a  mathematical  line  of  light,  and 
if  there  were  perfect  collinear  correspondence  between  object  and 
image,  the  spectrum  would  be  absolutely  pure,  and  the  image  of  the 
line-source  for  a  given  wave-length  of  light  would  be  itself  a  line 
occupying  a  perfectly  definite  and  distinct  position  in  this  ideal  spec- 
trum. 

Evidently,  the  purity  of  the  spectrum  will  depend  on  the  width  of 
the  slit-image  and  on  the  length  of  the  spectrum.  Let  8a  denote  the 
apparent  size  of  the  slit  as  viewed  from  the  first  face  of  the  prism,  and, 
similarly,  let  da'  denote  the  apparent  size  of  the  slit-image  for  light 
of  wave-length  X  as  viewed  from  the  last  refracting  plane.  The  greater 
the  dispersion  de/d\  of  the  light  of  wave-lengths  X  and  X  -f-  d\,  and 
the  smaller  the  magnitude  of  the  angular  width  da'  of  the  slit-image, 
the  greater  will  be  the  purity  of  the  spectrum  at  this  place  in  it;  and, 
hence,  as  a  measure  of  the  purity  of  the  spectrum,  HELMHOLTZ1  proposed 
that  we  employ  the  following  expression: 

P  =  fj  :  &«'•  (389) 

What  is  here  meant  by  the  image  of  the  slit  is  not  the  actual  or  "dif- 
fraction" image,  but  the  image  as  determined  on  the  assumption  of  the 
rectilinear  propagation  of  light  according  to  the  laws  of  Geometrical 
Optics. 

,  l  H.  VON  HELMHOLTZ:  Handbuch  der  physiologischen  Optik  (zweite  umgearbeitete 
Auflage,  Hamburg  u.  Leipzig,  1886),  p.  294.  In  regard  to  this  subject  see  also: 

H.  KAYSER:  Handbuch  der  Spectroscopie,  Bd.  I  (Leipzig,  1900),  pages  305  &  foil,  and 
pages  548  &  foil.; 

S.  CZAPSKI:  Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  pages  148 
&  foil.;  and 

F.  LOEWE'S  "  Die  Prismen  und  die  Prismensysteme  "  in  Die  Theorie  der  optischen 
Instrumente,  Bd.  i  (Berlin,  1904),  edited  by  M.  VON  ROHR,  pages  448  &  foil. 


494  Geometrical  Optics,  Chapter  XIII.  [  §  337. 

If,  therefore,  we  leave  out  of  account  the  diffraction-effects,  then, 
according  to  formula  (49)  of  §  97,  the  angular  width  of  the  slit-image 
formed  by  a  system  of  prisms,  which  is  composed  of  m  plane  refract- 
ing surfaces  and  in  which  the  first  and  last  media  are  both  air  (^  = 
nm  —  i),  is  given  by  the  following  formula: 

(390) 

where  8^  denotes  the  angular  width  of  the  slit  itself.  Accordingly, 
assuming  that  there  is  no  initial  dispersion,  and  employing  therefore 
formulae  (377),  we  find  for  the  purity  of  the  spectrum  produced  by  a 
system  of  prisms,  as  it  is  defined  by  equation  (389),  the  following 
expression  : 

P-    x      —        I       I       y 
•    rf\~  •  8a'm  "  rf\~  '  ^  =  t^ 

wherein  the  term  cos  a'Q  must  be  put  equal  to  unity  always.  Thus,  we  see 
that  the  magnitude  P  depends  not  merely  on  the  properties  of  the 
prism-system  but  on  the  width  of  the  slit  itself;  and,  hence,  the  purity 
of  the  spectrum,  as  defined  by  HELMHOLTZ,  is  not  by  itself  a  sufficient 
criterion  for  the  comparison  of  the  spectra  produced  by  different 
prism-systems. 

337.  Purity  of  Spectrum  in  Case  of  a  Single  Prism.  Consider  the 
spectrum  of  a  single  prism  surrounded  by  air.  In  this  special  case  let 
us  write  according  to  our  custom  : 


According  to  formula  (375),  we  have  here: 

Xl  =  sin  a['dn,    X2  =  —  sin  az'dn\ 

and,  hence,  putting  m  =  2  in  formula  (391),  we  obtain  for  the  purity 
of  the  spectrum  of  a  single  prism  in  the  region  corresponding  to  the 
light  of  wave-length  X: 

_         sin  /8         dn    I  , 

cos  «!  •  cos  a2  d\  dott  ' 

where  /3  =  o^  —  a.2  denotes  the  refracting  angle  of  the  prism.  We  see, 
therefore,  that  the  purity  of  the  spectrum  of  a  single  prism  surrounded 
by  air  is  proportional  to  the  so-called  ''characteristic  dispersion" 
(§  331)  of  the  prism-medium  and  is  inversely  proportional  to  the  width 
of  the  slit.  The  purity  of  the  spectrum  varies  also  with  the  angle  of 


§  338.]  Colour-Phenomena.  495 

incidence  (c^),  and  when  the  prism  is  adjusted  so  that  the  incident 
ray  "grazes"  the  first  face  (aL  =  90°),  we  find  P  =  oo.  The  advant- 
age of  using  the  prism  in  this  position  is  enormously  discounted, 
however,  on  account  of  the  great  loss  of  light  by  reflexion.  As  the 
angle  a^  decreases  from  the  value  a^  =  90°,  the  purity  P  diminishes 
also  until  it  attains  a  minimum  value  determined  by  that  value  of  the 
angle  ^  for  which  the  function  cosc^-cosc^  is  a  maximum.  This 
value  of  al  may  be  found  by  a  process  entirely  analogous  to  that 
employed  by  THOLLON  in  ascertaining  the  position  of  minimum  dis- 
persion, which  was  alluded  to  in  §  332;  in  fact,  by  merely  interchang- 
ing the  symbols  a{  and  a2  in  formula  (379),  we  obtain  immediately 
the  following  relation: 

<*2  =  ~  n2<*'i',  (393) 

which  gives  approximately  the  position  of  the  prism  for  minimum 
purity  of  the  spectrum  in  the  region  corresponding  to  the  light  of 
wave-length  X. 

If  the  prism  is  adjusted  in  the  position  of  minimum  deviation  for 
the  rays  of  wave-length  X  (which,  in  addition  to  other  advantages, 
is  also  the  position  in  which  the  loss  of  light  by  reflexion  is  least), 
we  must  introduce  in  formula  (392)  the  following  relations: 


whereby  we  obtain  : 

2  dn  i 

P  =  ~  tan  a,  —  ;  —  ; 
n  l  d\  &*! 

a  result  which  may  be  derived  also  directly  from  formula  (389)  by 
merely  remarking  that  for  the  position  of  minimum  deviation  we  have 
(see  §  86)  da'  =  da'2  =  da^  whereas  the  value  of  de/d\  is  given  here 
by  formula  (380).  The  purity  of  this  part  of  the  spectrum  depends 
only  on  the  refractive  index  n,  the  width  of  the  slit  and  the  form  of 
the  prism.  It  is  called  the  "normal  purity". 

338.  Diffraction-Image  of  the  Slit.  The  methods  of  Geometrical 
Optics  alone  are  not  sufficient  to  enable  us  to  ascertain  the  character  of 
the  slit-image  ;  this  problem  involves  not  merely  the  theory  of  refraction 
but  the  theory  of  diffraction  also.  According  to  this  latter  theory,  the 
image  of  a  luminous  line  (or  very  narrow  rectangular  aperture)  parallel 
to  the  edge  of  the  prism  is  never  itself  a  line,  but  a  far  more  complicated 
effect  which  we  have  not  space  to  investigate  here,  especially  too  as  a 
complete  exposition  of  the  matter  can  be  found  in  almost  any  standard 
work  on  Physical  Optics.  In  the  accompanying  diagram  (Fig.  150)  the 


496 


Geometrical  Optics,  Chapter  XIII. 


[  §  338. 


plane  of  the  paper  represents  a  principal  section  of  the  prism-system,  of 
which  the  traces  in  this  plane  of  the  first  and  last  surfaces,  /^  and  /*„/*„, 
are  shown  in  the  figure.  The  source  of  the  light  is  supposed  to  be  a 
small  luminous  line  perpendicular  at  6"  to  the  plane  of  the  paper.  The 
rays  emanating  from  this  line-source  are  made  parallel  by  a  "colli- 
mating"  lens;  so  that  the  straight  line  PQ  =  b±  is  the  trace  in  the 
plane  of  the  paper  of  the  portion  of  the  plane-wave  which  is  due  to 


Prism  Sjafem 


Convex  Lefts 


Central  Banal 
forA  +  d*. 


Central  Band 
iorJL 


Scree* 


FIG.  150. 

RESOLUTION  OF  I^INES  IN  PRISMATIC  SPECTRUM.  The  plane  of  the  paper  represents  the  plane 
of  a  principal  section  of  the  prism-system.  The  source  of  light  is  a  small  luminous  line  perpendic- 
ular to  plane  of  paper  at  the  point  marked  S.  The  straight  lines  MiMi  and  Mm^n*  are  the  traces  in 
the  plane  of  the  paper  of  the  first  and  last  refracting  planes,  respectively.  PQ  =  61  =  width  of  beam 
of  parallel  incident  rays ;  P'  Q1  =  bm  =  width  of  beam  of  parallel  emergent  rays  of  wave-length  A. 
Z  S'OS"  =  Se  =  angular  distance  of  slit-images  corresponding  to  light  of  wave-lengths  A  and  A  +  rfA. 

arrive  later  at  the  first  surface  MiMi  of  the  prism-system.  The  straight 
line  P'Q'  =  b'm  shows  the  trace  in  this  plane  also  of  the  corresponding 
emergent  plane-wave  for  light  of  wave-length  X.  A  convex  lens  inter- 
posed in  the  path  of  the  beam  of  emergent  rays  will  produce  on  a  screen 
situated  in  the  focal  plane  of  the  lens  an  image  of  the  slit.  If  this 
image  is  investigated  by  the  methods  of  Physical  Optics,  we  find  that 
the  image  of  a  vertical  line  at  S  consists  mainly  of  a  so-called  "central 
band"  of  light  of  a  certain  finite  horizontal  width  (which  depends  on 
the  focal  length  of  the  lens,  for  one  thing)  and  of  maximum  brightness 
along  a  vertical  line  perpendicular  to  the  plane  of  the  paper  at  the 
point  designated  in  the  diagram  by  5'.  On  either  side  of  this  vertical 
median  line  the  brightness  of  the  central  band  diminishes  very  rapidly 
to  absolute  darkness.  There  is  also  a  series  of  much  fainter  bands 
situated  symmetrically  on  both  sides  of  the  central  band,  but  for  all 
practical  purposes  the  central  band  alone  may  be  considered  as  the 
actual  and  effective  image  of  the  very  narrow  rectangular  aperture 
at  5. 

If  the  slit  is  illuminated  by  light  of  wave-lengths  X  and  X  +  d\, 


§  339.1  Colour-Phenomena.  497 

the  rectangular  beam  of  parallel  incident  rays  will  be  resolved  by  the 
prism-system  into  two  such  beams,  one  for  each  of  the  two  colours. 
Thus,  we  shall  have  at  the  points  S'  and  S"  on  the  screen  the  maxima 
of  brightness  of  the  two  slit-images  corresponding  to  the  light  of  wave- 
lengths X  and  X  +  d\.  Now,  in  order  that  these  images  whose  central 
portions  are  at  Sf  and  S"  may  be  far  enough  apart  to  be  distinguished 
by  the  eye  as  separate  and  distinct  images,  Lord  RAYLEIGH1  has  shown 
that  the  /.S'OS"  =  Semust  beat  least  equal  to  \/b'm,  where  b'm  =  P'Q' 
is  the  width  of  the  beam  of  emergent  rays  of  wave-length  X.  The 
width  of  the  beam  of  emergent  rays  will  depend  on  the  orientation  of 
the  prism-system,  as  is  evident  from  formula  (387),  and  the  angular 
interval  5e  of  the  centres  of  the  two  images  will  depend  on  this  also. 
In  order,  therefore,  to  resolve  a  "double  line'1  of  wave-lengths  X  and 
X  +  dX,  it  is  necessary  that  the  angular  interval  5e  shall  have  the 
following  value  at  least  : 

X  *ff  cosak 

k**!!—  >•  (394) 

i  &=i  *"*~'>3  ™jg 

In  the  special  case  when  the  rays  corresponding  to  the  light  of  wave- 
length X  traverse  the  prism-system  with  minimum  deviation,  we  have, 
according  to  the  formula  at  the  end  of  §  94: 


i  cos    k 
and  hence  the  condition  (394)  becomes  in  this  special  case: 

*-*-•  • 

#1 

In  the  case  of  a  single  prism  (m  =  2),  formula  (394)  is  as  follows: 

X    cos  <x{  •  cos  or2 

(39S) 


339.  Ideal  Purity  of  Spectrum.  According  to  RAYLEIGH'S  investi- 
gations, the  least  value  of  the  angular  interval  5e  necessary  in  order  to 
resolve  a  "double  line"  is  equal  to  half  the  width  of  the  central  band 
of  the  diffraction-image  of  the  slit;  and,  hence,  on  the  supposition  that 
the  object  is  a  luminous  line,  the  methods  of  Physical  Optics  show  that 
the  angular  width  of  the  image  =  25e  =  2\fb'm.  HELMHOLTZ  defines  the 

1  Lord  RAYLEIGH:  Investigations  in  Optics:  Phil.  Mag.  (5)  viii  (1879),  pages  261-274, 
403-411,  477-486;  and  (5)  ix  (1880),  pages  40-55.  See  also  article  "Wave  Theory", 
9th  ed.  of  Encyclopedia  Britannica,  xxiv,  430-434. 

33 


498  Geometrical  Optics,  Chapter  XIII.  [  §  340. 

purity  of  the  spectrum  as  equal  to  the  ratio  of  the  dispersion  to  the 
angular  width  of  the  image  (§  336);  and,  hence,  the  so-called  Ideal 
Purity1  of  the  spectrum  (which  may  be  distinguished  by  the  symbol 
PQJ  where  the  zero-subscript  is  written  to  indicate  that  the  slit  in  this 
case  is  infinitely  narrow)  may  be  denned  by  the  following  equation : 

PO  =  ^|--  (396) 

Accordingly,  the  Ideal  Purity  of  Spectrum  produced  by  a  prism,  in  the 
case  when  the  slit  is  infinitely  narrow,  is  proportional  to  the  product  of 
the  dispersion  by  the  width  of  the  emergent  beam;  both  of  which  factors, 
as  above  remarked,  depend  on  the  orientation  of  the  prism  (or  prism- 
system). 

Thus,  in  the  case  of  a  single  prism,  we  obtain  by  formulae  (378), 
(387)  and  (396): 

bl  sin  j8        dn 

0      2X    cos  <*!  •  cos  az  d\ ' 

which  should  be  compared  with  formula  (392). 

If  s  and  /  denote  the  lengths  of  the  ray-paths  within  the  prism  of 
the  rays  of  wave-length  X  which  are  nearest  to  the  refracting  edge  of 
the  prism  and  farthest  from  it,  respectively  (see  Fig.  151),  it  is  easy 
to  show  that 

sin  ft 

1  cose*! «cos  a2' 
and,  hence,  we  may  write: 

t  —  s  dn  /     o\ 

Po==^T5x*  (398) 

340.  Resolving  Power  of  Prism-System.  The  magnitude  denoted 
above  by  the  symbol  P0  is  closely  related  to  what  Lord  RAYLEIGH 
calls  the  Resolving  Power  of  the  Prism-System.  Thus,  if  d\  is  the 
difference  of  wave-length  of  two  "lines"  of  the  spectrum  whose  mean 
wave-length  is  X  and  which  are  just  barely  separated  in  the  spec- 
trum, the  resolving  power  p  is  denned  by  RAYLEIGH  as  follows: 


RAYLEIGH'S  investigation  of  the  joint  effect  of  dispersion  and  width 
of  beam  on  the  resolving  power  of  a  prism  is  as  follows : 

*See  H.  KAYSER:   Handbuch  der  Spectroscopie,  Bd.  I    (Leipzig,  1900),  p.  307.     See 
also  F.  L.  O.  WADSWORTH:  The  Modern  Spectroscope:  Aslrophys.  Journ.,  i  (1895),  52-79. 


§  340.]  Colour-Phenomena.  499 

The  straight  line  PQ  (Fig.  151)  represents  the  trace  in  the  plane  of 
a  principal  section  (plane  of  the  paper)  of  the  plane  wave-front  of  the 
light  at  some  instant  prior  to  its  arrival  at  the  first  face  of  the  prism ; 
and  the  straight  line  P'Q'  represents  in  the  same  way  the  position  of 
the  wave-front  of  the  light  of  wave-length  X  at  some  subsequent  instant 
after  the  waves  have  traversed  the 
prism.  Similarly,  also,  the  straight  line 
P"Q"  represents  at  this  same  later 
instant  the  position  of  the  plane  wave- 
front  for  the  light  of  wave-length  X  + 
d\.  As  a  matter  of  fact,  the  two  rays 
of  wave-lengths  X  and  X  +  d\,  which 
meet  the  first  face  of  the  prism  at  the 
same  point,  will  thereafter  pursue  FIG.ISI. 

slightly  different  geometrical    paths;          RESOLVINO  POWER  OF  A  PRISM. 
but  by  virtue  of  FERMAT'S  Minimum 

Principle  (Art.  n),  this  difference  will  be  entirely  negligible  in  com- 
parison with  the  actual  distances  traversed  by  the  rays;  and,  hence, 
we  may  consider  that  the  rays  pursue  the  same  routes  both  within 
and  without  the  prism,  as  represented  in  the  diagram.  Thus,  for 
example,  if  s  and  /  denote  the  lengths  of  the  ray-paths  within  the 
prism  of  the  rays  of  wave-length  X  which  are  nearest  to  the  refracting 
edge  and  farthest  from  it,  respectively,  these  same  magnitudes  will 
denote  also  the  lengths  of  the  ray-paths  within  the  prism  of  the  corre- 
sponding pair  of  rays  for  light  of  wave-length  X  -f  d\.  The  refractive 
indices  of  the  prism  for  light  of  wave-lengths  X  and  X  -f-  d\  will  be 
denoted  by  n  and  n  +  dn,  respectively.  Finally,  the  prism  is  supposed 
to  be  surrounded  on  both  sides  by  air  whose  dispersion  is  so  slight  as 
to  be  negligible. 

According  to  Art.  n,  the  optical  lengths  of  the  paths  from  P  to  P' 
and  from  Q  to  Qr  for  light  of  wave-length  X  are  equal ;  as  is  true,  like- 
wise, with  respect  to  the  optical  lengths  of  the  paths  from  Q  to  Q" 
and  from  P  to  P"  for  light  of  wave-length  X  +  d\.  Evidently,  the 
optical  length  of  the  path  from  Q  to  Q"  for  the  light  of  wave-length 
X  -f  d\  is  longer  than  that  from  Q  to  Q'  for  the  light  of  wave-length 
X  by  the  amount  s-dn  +  (?'(?";  and,  in  the  same  way,  the  optical 
length  of  the  path  from  P  to  P"  for  the  light  of  wave-length  X  +  d\ 
exceeds  that  from  P  to  P'  for  the  light  of  wave-length  X  by  the  amount 
t-dn  —  P"P'\  and  since  these  excesses  must  be  equal,  we  find: 

(/  -  s)-dn  =  P"P'  +  Q'Q"  =  *P"P'. 


500  Geometrical  Optics,  Chapter  XIII.  [  §  340, 

Now  Z  P"OPf  =  de  is  the  angular  interval  between  the  maxima  of 
intensity  of  the  diffraction-images  corresponding  to  the  light  of  wave- 
lengths X  and  X  -f-  d\  ;  and 


—  2 


P'O'       "  P'Q'  b' 

where  b'  =  P'Q'  is  the  width  of  the  beam  of  emergent  rays  of  wave- 
length X.     Consequently,  we  have  here: 

t-s  , 


But,  according  to  RAYLEIGH'S  conclusion,  the  least  angle  subtended 
by  a  "double  line"  of  mean  wave-length  X  must  be  equal  to  \/b'  in 
order  that  it  may  be  fairly  resolved  in  the  spectrum  (§  338)  ;  that  is, 
the  very  smallest  value  of  the  angle  de  under  these  circumstances  is: 


and,  consequently,  RAYLEIGH  finds: 

X  =  (/  -  s)-dn, 
or 

X  .dn  .      . 

*-*x-<'-j)dx'  (400) 

and,  hence,  referring  to  formula  (398),  we  see  that  we  have  the  follow- 
ing relation  between  the  Ideal  Purity  P0  of  the  spectrum  and  the 
Resolving  Power  p  of  the  prism  : 

P  =  2XP0.  (401) 

According  to  formula  (400),  the  Resolving  Power  of  a  prism  is  propor- 
tional to  the  product  of  the  difference  of  the  lengths  (t  —  s)  of  the 
ray-paths  within  the  prism  of  the  two  extreme  rays  of  the  pencil  and 
the  characteristic  dispersion  dn/d\  of  the  dispersive  material  of  the 
prism.  If,  for  example,  the  value  of  the  difference  (/  —  s)  is  the  same 
for  two  prisms  of  the  same  material,  the  Resolving  Power  of  both  prisms 
will  be  the  same,  although  the  prisms  may  have  different  refracting 
angles  and  may  be  oriented  differently.  It  is  assumed  only  that  the 
slit  is  infinitely  narrow. 

If  one  of  the  extreme  rays  passes  through  the  prism-edge,  then 


§  341.]  Colour-Phenomena. 

5  =  0,  and  formula  (400)  reduces  to  the  following: 

dn 


501 


(402) 


FIG.  152. 


If  the  prism  is  adjusted  in  the  position  of  minimum  deviation,  and  if 
the  entire  extent  of  the  prism  is  traversed  by  the 
rays,  the  Resolving  Power  of  the  prism  depends  only 
on  the  thickness  of  the  prism  at  the  base. 

The  formulae  obtained  above  may  be  extended  at 
once  to  a  system  of  glass  prisms  separated  from 
each  other  by  air,  provided  the  glass  prisms  are 
all  made  of  the  same  kind  of  glass.  For  such  a 
system,  (/  —  s)  in  formula  (400)  will  denote  the 
difference  in  aggregate  thickness  of  the  dispersive 
material  through  which  the  extreme  rays  of  the 
pencil  have  passed.  If  the  prisms  are  all  adjusted 
so  that  the  rays  traverse  them  symmetrically,  and  if  the  upper  extreme 
ray  passes  through  the  edges  of  all  the  prisms,  then  /  in  formula 
(402)  denotes  the  sum  of  the  "bases"  of  the  prisms. 

SCHUSTER1  remarks  that  "the  resolving  power  of  prisms  depends  on 
the  total  thickness  of  glass,  and  not  on  the  number  of  prisms,  one 
large  prism  being  as  good  as  several  small  ones".  Thus,  all  the  prisms 
shown  in  Fig.  152  "would  have  the  same  resolving  power,  though  they 
would  show  very  considerable  differences  in  dispersion". 

341.  According  to  CAUCHY'S  Dispersion-Formula  (see  §  327),  we 
may  write  approximately  : 

n  =  A  +  B\~2', 

and,  hence,  by  formula  (402)  the  resolving  power  of  a  prism  of  base 
/is: 

p=-2B~.  (403) 

We  may  say,  therefore,  roughly  speaking,  that  the  Resolving  Power  of 
a  prism  is  inversely  proportional  to  the  cube  of  the  wave-length  ;  and 
hence  the  Resolving  Power  is  much  greater  for  light  of  short  wave- 
lengths. The  Resolving  Power  of  a  grating  is  the  same  for  all  wave- 
lengths, and  hence  a  grating-spectroscope  is  not  so  good  as  a  prism- 
spectroscope  for  resolving  the  ultra-violet  lines  of  the  spectrum. 

The  value  of  the  co-efficient  B  in  formula  (403)  depends  on  the 

1  A.  SCHUSTER:  An  Introduction  to  the  Theory  of  Optics  (London,  1904),  p.  144. 


502  Geometrical  Optics,  Chapter  XIII.  [  §  342. 

material  of  the  prism.  Lord  RAYLEIGH  gives  the  following  calculation 
of  tfie  thickness  /  of  a  prism  made  of  the  "extra  dense  flint"  glass  of 
Messrs.  CHANCE  Bros,  that  is  necessary  in  order  to  resolve  the 
FRAUNHOFER  double  D-line.  The  indices  of  refraction  of  this  glass 
for  light  corresponding  to  the  FRAUNHOFER  lines  C  and  D  are : 

nc=  1.644866,     nD  =  1.650388; 
and  the  wave-lengths  in  centimetres  are: 

Xc  =  6.562  -iQ-5,     Xz>  =  5.889 -io~5. 
Thus,  we  find: 

B  =  -=2 ~*  —  0.984 -io~10. 

Now 

X3        X     X3  X4 


*2B      d\  2B      2B-d\ 

For  theZMine:  X  =  5.889 -io~5,  d\  =  0.006 -io~5  (difference  between 
Dl  and  Z>2). 

Accordingly,  we  find  /  =  1.02  cm.,  which  is,  therefore,  the  necessary 
thickness  of  a  prism  of  this  material  in  order  to  resolve  the  double 
ZMine.  Moreover,  Lord  RAYLEIGH,  testing  this  result  by  experiment, 
found  that,  as  a  matter  of  fact,  a  prism-thickness  of  between  1.2  and 
1.4  cm.  was  needed  for  this  purpose. 

342.  The  Resolving  Power  of  a  system  of  prisms  of  different 
materials  is  given  by  the  following  formula : 

X      *=™^1  dnk 

P  =  -jT  =    L^i    ftfe  ~  sk)  ~JJT  I  (4°4) 

where  sk  and  tk  denote  the  lengths  of  the  ray-paths  of  the  extreme  rays 
between  the  &th  and  the  (k  +  i)th  plane  refracting  surfaces. 

For  example,  in  an  AMICI  Direct-Vision  Prism  (§  335),  consisting 
of  two  prisms  of  crown  glass  cemented  to  a  prism  of  flint  glass,  as 
represented  in  Fig.  153,  we  have: 

and  hence  by  formula  (404) : 

dn\  dn'2 

In  this  combination  the  dispersion  of  the  crown  glass  is  opposed  to 


§  343.] 


Colour-Phenomena. 


503 


FIG.  153. 
RESOLVING  POWER  OF  AMICI  "  DIRECT- VISION  "  PRISM. 

that  of  the  flint  glass,  and  the  Resolving  Power  of  the  system  is  not 
great. 

II.    THE  CHROMATIC  ABERRATIONS. 

ART.  108.     THE    DIFFERENT   KINDS   OF   ACHROMATISM. 

343.  When  a  ray  of  white  light  is  incident  on  a  refracting  surface, 
it  will,  in  general,  be  resolved  at  the  point  of  incidence  into  a  pencil  of 
coloured  rays,  since,  as  we  have  seen,  the  index  of  refraction  depends 
on  the  colour  of  the  light.  Thus,  for  example,  if  P  designates  the 
position  of  a  radiant  point  emitting,  say,  red  and  blue  light,  and  if  Bl 
designates  the  position  of  a  point  on  the  first  surface  of  a  centered  sys- 
tem of  spherical  refracting  surfaces,  and,  finally  if  TT  designates  a  trans- 
versal plane  perpendicular  to  the  optical  axis,  then  corresponding  to 
an  incident  ray  proceeding  along  the  straight  line  PBlt  there  will  be 
a  red  image-ray  whicn  will  cross  the  plane  TT  (really  or  virtually)  at  a 
point  Pf  and  likewisealso  a  blue  image-ray  which  will  cross  the  plane  TT  at 
a  point  P',  which,  in  general,  will  be  different  from  the  point  P'.1  Since 
the  positions  of  the  focal  points  and  the  magnitudes  of  the  focal  lengths 
of  an  optical  system  depend  also  on  the  indices  of  refraction  of  the 
media  traversed  by  the  rays,  and  since  the  values  of  these  indices 
depend  on  the  colour  of  the  light,  it  is  evident  that  the  same  optical 
system  will  produce  as  many  coloured  images  of  a  given  object  as 
there  are  colours  in  the  light  emitted  by  the  object;  and,  in  general, 
also,  these  images  will  be  formed  at  different  places  and  will  be  of 
different  sizes.  The  entire  series  of  images  may  be  described  as  an 
image  affected  with  chromatic  aberrations.  Even  if  the  image  were 

'  *  In  the  following  pages  of  this  chapter,  whenever  we  have  to  deal  with  two  colours, 
the  letters  and  symbols  which  relate  to  the  second  colour  will  be  distinguished  from  the 
corresponding  letters  and  symbols  which  relate  to  the  first  colour  by  means  of  a  dash 
written  immediately  above  the  character.  It  is  true  this  same  method  of  notation  was 
used  in  the  theory  of  astigmatism  to  distinguish  between  the  meridian  and  sagittal  rays; 
but  no  confusion  is  likely  to  occur  on  this  account,  and  for  various  reasons  it  is  convenient 
to  use  this  same  device  here  in  a  new  sense. 


504  Geometrical  Optics,  Chapter  XIII.  [  §  343. 

otherwise  perfect  and  free  from  all  the  so-called  spherical  aberrations, 
the  definition  of  the  image  will  generally  be  seriously  impaired  on 
account  of  colour-dispersion  alone,  and  hence  one  of  the  most  important 
problems  of  practical  optics  is  to  correct,  as  far  as  possible,  the  chro- 
matic aberrations  and  to  produce  an  optical  system  that  is  more  or 
less  achromatic. 

The  problem  here  mentioned  is  still  further  complicated  by  the  fact 
that  not  only  are  the  fundamental  characteristics  of  the  optical  system 
(viz.,  the  positions  of  the  focal  points  and  the  magnitudes  of  the  focal 
lengths)  dependent  on  the  indices  of  refraction  of  each  medium,  but 
the  various  spherical  aberrations,  which  are  encountered  when  the 
rays  are  not  infinitely  near  to  the  optical  axis,  are  likewise  functions 
of  the  indices  of  refraction;  so  that  we  may  have  also  chromatic  varia- 
tions of  the  spherical  aberrations,  even  though  the  optical  system  has 
been  corrected  so  that  the  focal  points  and  the  focal  lengths  are  the 
same  for  all  wave-lengths  of  light.  As  a  matter  of  fact,  all  the  prop- 
erties of  a  centered  system  of  spherical  refracting  surfaces  are  depend- 
ent in  some  way  or  other  on  the  indices  of  refraction,  and  hence  they 
are  all  variable  with  the  colour  of  the  light.  The  term  "achromatism'* 
by  itself  is,  therefore,  entirely  indefinite,  for  the  system  may  be  achro- 
matic in  one  sense  and  not  at  all  so  in  other  senses.  For  example, 
the  images  corresponding  to  the  different  colours  may  all  be  formed 
at  the  same  place,  and  yet  be  of  different  sizes;  or  the  system  may  be 
achromatic  with  respect  to  Distortion  or  with  respect  to  the  Sine- 
Condition,  etc.,  and  at  the  same  time  affected  with  colour-dispersion 
in  a  variety  of  other  ways.  Obviously,  it  will  not  be  possible  to  correct 
all  these  different  kinds  of  chromatic  aberrations  at  the  same  time; 
and,  in  fact,  in  order  to  have  a  distinct  image  (which  is  the  primary 
aim  of  an  optical  instrument),  this  will  not  be  necessary,  as  some  of 
the  chromatic  aberrations  are  comparatively  unimportant,  depending 
on  the  purpose  which  the  apparatus  is  intended  to  fulfil. 

An  optical  system  which  produces  two  coloured  images  of  a  given 
object  at  the  same  place  and  of  the  same  size  is  said  to  be  completely 
achromatic  for  these  two  colours.  The  images  of  other  colours  will, 
however,  generally  be  different  as  to  both  size  and  position,  and  the 
effect  on  the  resultant  image  usually  appears  in  a  coloured  margin  or 
"secondary  spectrum"  (§  329). 

But  usually  the  best  we  can  do  is  to  contrive  to  obtain  a  partial 
achromatism  of  some  sort,  and  especially  one  of  the  two  following 
kinds:  achromatism  with  respect  to  place  (so  that  the  two  coloured 
images,  although  of  unequal  sizes,  are  both  formed  in  the  same  image- 


§  344.]  Colour-Phenomena.  505 

plane),  or  achromatism  as  to  magnification  (so  that  the  two  coloured 
images,  although  differently  situated,  are  of  equal  size).  In  many 
cases,  indeed,  it  will  be  found  quite  sufficient  to  effect  a  partial  achroma- 
tism of  one  or  other  of  these  two  kinds.  Thus,  for  example,  it  is 
essential  that  the  coloured  images  formed  by  the  objectives  of  tele- 
scopes and  microscopes  shall  be  situated  as  nearly  as  possible  at  the 
same  place;  whereas,  since  the  images  do  not  extend  far  from  the 
optical  axis,  the  unequal  colour-magnifications  are  comparatively  neg- 
ligible. On  the  other  hand,  in  the  case  of  the  eye-pieces  of  these 
instruments,  whose  particular  office  is  to  produce  extended  images  of 
the  small  images  formed  by  the  objectives,  the  main  point  is  to  ob- 
tain achromatism  as  to  magnification,  whereas  the  slight  differences 
in  the  distances  of  the  coloured  images  are  of  relatively  small  import- 
ance. As  a  rule,  it  may  be  stated  that  for  an  optical  system  which 
produces  a  real  image,  it  is  more  desirable  to  have  achromatism  with 
respect  to  the  place  of  the  image;  whereas  if  the  image  is  virtual, 
achromatism  with  respect  to  the  magnification  is  likely  to  be  the  more 
important  requirement. 

In  the  following  pages  it  is  proposed  to  develop  the  formulae  for  the 
numerical  calculation  of  the  more  important  of  the  chromatic  aber- 
rations, and  to  determine  the  conditions  that  are  necessary  in  order  to 
abolish  or  diminish  them.  In  this  investigation  it  will  be  assumed 
(except  in  the  brief  treatment,  at  the  end  of  the  chapter,  of  the  chro- 
matic variations  of  the  spherical  aberrations)  that  we  are  concerned 
only  with  paraxial  rays,  so  that  between  the  object  and  its  image  in 
any  one  definite  colour  there  is  complete  collinear  correspondence. 

As  to  notation,  let  us  state  here  that  the  change  of  a  magnitude  x 
in  consequence  of  a  finite  variation  of  the  wave-length  of  the  light 
from  the  value  X  to  the  value  X  will  be  indicated  by  the  capital  letter 
D  written  immediately  in  front  of  the  symbol  of  the  magnitude,  thus: 

Dx  =  x  —  x. 


ART.  109.     THE    CHROMATIC    VARIATIONS    OF    THE    POSITION    AND    SIZE 

OF    THE    IMAGE,    IN    TERMS    OF    THE    FOCAL    LENGTHS    AND 

FOCAL    DISTANCES    OF  THE    OPTICAL   SYSTEM. 

344.  Let  Al  and  Am  designate  the  positions  of  the  vertices  of  the 
first  and  last  surfaces,  respectively,  of  a  centered  system  of  m  spherical 
refracting  surfaces,  and  let  F,  E'  and  F,  Ef  designate  the  positions 
on  the  optical  axis  of  the  focal  points  of  the  system  for  the  two  colours 
corresponding  to  light  of  wave-lengths  X  and  X,  respectively.  At  a 


506  Geometrical  Optics,  Chapter  XIII.  [  §  345. 

point  M  on  the  optical  axis,  erect  the  perpendicular  MQ\  and  let 
M'Q'  and  M'Q'  be  the  GAUssian  images  in  the  two  colours  correspond- 
ing to  the  object  MQ.  We  shall  employ  here  the  following  symbols: 

z  =  AiF,        z  =  AJ?,  Dz  =  z  -  z  =  FF, 

z'  =  AmE',  z'  =  AmE',  Dz'  =  z'-z'  =  E'E', 

u  =  A}M,  u'  =  AmM',      u'  =  AmM',     Du'  =  u'  -  u1  =  M'M', 

x  =  FM,  x'  =  E'M',        x  =  FM,          x'  =  E'M', 

y  =  MQ,  y'  =  M'Q',       y'  =  M'Q'. 

Denoting  the  focal  lengths  of  the  system  for  the  two  colours  by 
/,  e'  and  /,  e',  we  have  the  following  set  of  equations  (§  178): 

xx'=fe',  xx'  =fe't 


- 

X     —  -  •  JL        -  -     —      t 

y       x  y       x 

where  Y,  Y  denote  the   lateral   magnifications  for  the  two  colours. 
Now 

x  =  x  -  Dz,    x'  =  x'  +  Du'  -  Dz'-, 

and,  hence,  eliminating  x  and  x'  and  solving  for  Du',  we  obtain  from 
the  upper  pair  of  equations  (405)  : 

*,-*  +  *•*+£">.  (406) 

Similarly,  eliminating  x  from  the  lower  pair  of  equations  (405),  we 
obtain  : 


These  Difference-Formulae,  which  are  given  by  KoENiG,1  give  the 
variations  of  the  position  and  magnification  of  the  image  of  an  object 
corresponding  to  any  arbitrary  variations  of  the  fundamental  charact- 
eristics of  the  optical  system. 

345.     We  may  consider  several  special  cases  as  follows: 
(i)  If  the  optical  system  is  achromatic  with  respect  to  the  position 
of  the  image,  then  we  shall  have  Du'  =  o,  and  from  equation  (406)  we 
obtain  in  this  case  the  following  quadratic  equation  with  respect  to  x  : 

xz-Dz'+  {D(fe')  -Dz-Dz'}x+fe'-Dz  =  o; 

1  See  A.  KOENIG:  "  Die  Theorie  der  chromatischen  Aberrationen  ",  Chapter  VI  of 
Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR. 
See  p.  345. 


§  345.]  Colour-Phenomena.  507 

so  that,  in  general,  there  will  be  two  positions  of  the  object  for  which 
its  images  in  the  two  given  colours  will  be  formed  in  the  same  trans- 
versal plane;  but  if  the  roots  of  the  quadratic  are  imaginary,  there 
will  be  no  position  of  the  object  for  which  the  system  can  have  this 
kind  of  achromatism.  In  the  special  case  when 

Dz  =  Dz'  =  D(fe')  =  o, 

the  quadratic  equation  will  be  satisfied  for  all  values  of  x;  and  we 
have  then  what  is  sometimes  called  stable  achromatism  with  respect 
to  the  place  of  the  image.  If,  also,  the  first  and  last  media  are  ident- 
ical, we  have  DY  =  o*and  the  system  will  be  completely  achromatic, 
in  the  sense  in  which  this  term  was  defined  in  §  343. 

(2)  If  the  system  is  achromatic  with  respect  to  the  lateral  magni- 
fication, then  DY  =  o.  This  condition  is  satisfied  by  x  =  x  =  oo, 
Y  =  F  =  o;  and,  also,  according  to  equation  (407),  by  the  following 
value  of  x: 


If  Df  =  Dz  =  o,  then  DY  =  o  for  all  values  of  x. 

If  g,  g  denote  the  ordinates  of  the  points  where  an  incident  ray 
emanating  from  the  axial  object-point  M  crosses  the  primary  focal 
planes  corresponding  to  the  two  colours,  then 

x  :x  =  g  :g; 
and  if  the  two  magnifications  F,  F  are  equal,  then 

*:*=/:/; 


and  hence: 

and,  since  (§  178) 


g  =  e'-tan0',     g  =  e'-tan  0', 


where  0',  0"'  denote  the  slopes  of  the  pair  of  coloured  image-rays  cor- 
responding to  the  given  incident  ray,  we  have  : 

e'-tan0'  :e'-tan  H'  =/  :/. 

If,  therefore,  the  first  and  last  media  are  identical,  sojhat  e'  =  —  /, 
e'  =  —  /,  we  obtain  0'  =  "0';  and  hence  when  F  =  F,  and  n  =  n', 
n  =  n',  the  pair  of  coloured  emergent  rays  corresponding  to  a  given  inci- 
dent ray  will  be  parallel. 


508  Geometrical  Optics,  Chapter  XIII.  [  §  346. 

• 

(3)  If  the  optical  system  is  achromatic  with  respect  to  the  positions 
of  the  Focal  Points,  then  Dz  =  Dzf  =  o,  and  we  have : 

Du'  =  ~72~xf  =  2—x',  approximately; 

which  shows  that  the  variation  Du'  of  the  position  of  the  image  will 
depend  on  the  variation  of  the  Focal  Length.     If  Df  =  o,  or  if  x'  =  o, 
the  two  coloured  images  of  the  given  object  will  lie  in  the  same  plane 
(Du'  =  o). 
Moreover,  we  have  here : 

f,     or    ly-f/; 

and  hence  the  variation  Dx'  in  the  size  of  the  image  is  independent  of 
the  value  of  x.  Accordingly,  there  is  no  value  of  x  that  will  make 
Dyr  =  o;  except  the  value  x  =  oo,  for  which  we  have  yr  =  y'  =  o. 
Since,  however,  Dxf  is  proportional  to  y',  the  variation  Dyr  will  be 
more  and  more  nearly  negligible,  the  smaller  y'  is;  and  hence  this 
variation  will  be  very  slight  in  the  central  part  of  the  field  of  view. 

(4)  Finally,  if  the  system  is  achromatic  with  respect  to  the  Focal 
Lengths  (/  =  /,  e'  =  ef),  the  positions  of  the  Focal  Points  Ft  F  and 
£',  E'  will,  in  general,  be  different.     Putting  Df  =  Der  =  o  in  formulae 
(406)  and  (407),  we  obtain: 

Dz  ,  Du'  -  Dz'  f            ,                            Du'  -  Dz' 
—  +  -  -j2--(x-Dz)  =o,    DY= . 

ART.    110.     FORMULA   ADAPTED   TO   THE    NUMERICAL    CALCULATION   OF 
THE  CHROMATIC  VARIATIONS   OF  THE   POSITION  AND    MAGNIFICA- 
TIONS OF  THE  IMAGE    OF  A   GIVEN    OBJECT   IN  A  CENTERED 
SYSTEM   OF   SPHERICAL  REFRACTING   SURFACES. 

346.     Chromatic  Longitudinal  Aberration. 

The  relation  between  the  intercepts  u,  u'  of  a  paraxial  ray  before 
and  after  refraction,  respectively,  at  a  spherical  surface,  of  radius  r, 
which  separates  media  whose  indices  of  refraction  for  light  of  wave- 
length X  are  denoted  by  n  and  nr  is  given  by  the  following  equation: 


(i        i\        ,/i       i\        7 
n\ 1=  n'[ ,  ]  =  /; 

\r       u)          \r     u1 ) 


and,  hence,  after  elimination  of  r,  we  obtain: 

Dn      n-Du       r  Dnr      n'-Du' 
DJ  =  J — +-      -  =  J  —r  +      ,  ., 

n          uu  n'          u'u' 


§  346.]  Colour-Phenomena.  509 

where  the  symbols  with  dashes  above  them  relate  to  the  same  incident 
ray  for  light  of  wave-length  X.     Accordingly,  we  find: 


uu 

where,  according  to  ABBE'S  system  of  notation,  the  operator  A,  written 
before  an  expression,  indicates,  as  always  heretofore,  the  difference 
of  the  values  of  the  expression  before  and  after  refraction.  Thus,  for 
the  kth  surface  of  a  centered  system  of  spherical  refracting  surfaces, 
we  have: 

J*^\_jfDn\ 

\    uu    )k  \n  )k 

Since  the  distance  measured  along  the  optical  axis  between  the  kth 
and  the  (k  -f-  i)th  spherical  surfaces  is 


we  have: 

Duk+l  =  Duk. 

Moreover,  if  hk,  Ek  denote  the  incidence-heights,  for  the  rays  of  the 
two  colours,  at  the  kth  surface,  then  : 


Taking  note  of  these  relations,  and  multiplying  both  sides  of  equation 
(408)  by  h,.-hft,  and  then  giving  k  in  succession  all  integral  values 
from  k  =  i  to  k  =  m,  and  adding  together  all  the  equations  thus 
formed,  we  obtain: 


and  if  the  object  is  without  dispersion,  that  is,  if  Du^  =  o,  we  derive 
finally  the  following  formula  for  the  so-called  chromatic  longitudinal 
aberration1  of  a  centered  system  of  m  spherical  surfaces  for  a  given 
position  of  the  axial  object-point: 

(409) 


1  See  A.  KOENIG:  Die  Theorie  der  chromatischen  Aberrationen  :  Chapter  VI  of  Die 
Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904).  edited  by  M.  VON  ROHR.  See 
page  341. 


510  Geometrical  Optics,  Chapter  XIII.  [  §  347. 

The  chromatic  lateral  aberration,  with  respect  to  a  given  axial  object- 
point,  is  defined  by  ABBE  as  the  radius  of  the  cross-section  of  the 
bundle  of  image-rays  of  the  second  colour  made  by  the  image-plane 
of  the  first  colour.  Its  magnitude  is: 


and  we  can  obtain  here  also,  exactly  as  in  §  266,  a  more  or  less  artificial 
measure  of  the  "indistinctness  of  the  image"  due  to  the  chromatic 
aberration. 

The  condition  that  the  optical  system  shall  be  achromatic  with 
respect  to  the  place  of  the  image  of  a  given  object  is: 

(410) 

347.    Differential  Formulae  for  the  Chromatic  Variations. 

If  we  assume  that  the  variation  of  n  is  infinitely  small,  we  shall 
obtain  for  the  chromatic  variations  of  the  place  and  magnifications 
of  the  image  a  series  of  differential  formulae,  which  are  much  simpler 
than  the  corresponding  difference-formulae,  and  which  are  also  suf- 
ficiently accurate  in  most  cases  even  for  a  finite  colour-interval.  Thus, 
for  example,  by  differentiation,  we  easily  derive  the  following  formula1 
for  the  Chromatic  Longitudinal  Aberration  : 

In  a  centered  system  of  m  spherical  surfaces,  the  Lateral  Magnifica- 
tion is  (§138): 

F-£ff?;  (412) 

nin  &=i  uk 
the  Angular  Magnification  is  (cf.  §  179  and  §  193): 


&=m 


and  the  Axial  Magnification  is: 


aSee  L.  SEIDEL:   Zur  Theorie  der  Fernrohr-Objective:    Astr.  Nachr.,  xxxv  (1853), 
No.  835,  pages  301-316. 


§347.] 


Colour-Phenomena. 


511 


By  differentiation,  we  obtain : 

Y  ==^~  " 

dZ          *S 

~Z  =  ""  h  ' 


du 


Since 

we  have  evidently : 


dX      dni  _  dn'm 
X  "-  n,  '•  nm 


and     dt 


k-l 


and  if  we  assume  that  the  object  is  without  dispersion  (dut  =  o),  we 
may,  therefore,  write  formulae  (415)  as  follows: 


n. 


dY  _  dn 

~Y""^ 

dZ=_dui 
Z  u'^ 


^    d 


k-l 


u. 


lk-l 


dX      dn±      dnm        dun 

T  =  ^T  "ic+2"Z" 


(4l6) 


which  forms  are  more  convenient  than  equations  (415)  in  case  we  have 
to  determine  the  chromatic  variations  of  the  magnification  for  the 
special  case  when  du'm  =  o. 

If  the  first  and  last  media  are  identical  (n^  =  n'm),  we  have,  ac- 
cording to  formulae  (416): 


dY 
Y 


a  X 


um 


du: 


k-l* 


and  the  condition  that  dZ  =  o  is  also  in  this  case  identical  with  the 
conditions  that  dY  and  dX  shall  vanish.  Under  these  circumstances 
(see  §  345),  the  pair  of  coloured  emergent  rays  corresponding  to  a  given 
incident  ray  will  be  parallel.  If,  moreover,  the  distances  (d)  between 
each  pair  of  successive  surfaces  are  all  vanishingly  small  (system 
of  infinitely  thin  lenses  in  contact),  the  condition 

dX  =  dY  =  dZ  =  o 
is  identical  with  the  condition  dum  =  o. 


512 


Geometrical  Optics,  Chapter  XIII. 


[  §  348. 


KoENiG,1  in  his  excellent  treatise  on  the  Chromatic  Aberrations, 
derives  also  the  formulae  for  DX/X,  DY/Y,  and  DZ/Z]  but  these 
difference-formulae,  although  easily  obtained,  are  rather  complicated 
in  form. 

348.  KOENIG  remarks  also  that,  in  the  case  of  optical  systems 
affected  with  chromatic  longitudinal  aberration,  instead  of  trying  to 

abolish  the  variation  of  the  chro- 
matic lateral  magnification,  it  is 
sometimes  preferable  to  adjust  the 
transversal  focussing  plane  in  the 
position  M"Q"  (Fig.  154),  where  Q" 
designates  the  point  of  intersection 
of  the  chief  image-rays  of  the 
two  bundles  of  coloured  rays  corre- 
sponding to  the  bundle  of  incident 
rays  emanating  from  the  extra- 
axial  object-point  Q\  so  that  the 
projections  on  the  focussing  plane 
of  the  two  coloured  images  M'Q' 
and  M'~Q'  of  the  object  MQ  are  of 

the  same  size,  viz.,  M"Q".  If  M',  M'  designate  the  positions  of  the 
points  where  the  chief  rays  of  the  bundles  of  image-rays  of  the  two 
colours  cross  the  optical  axis,  then  evidently,  as  the  diagram  shows: 


'  M'M' 


FIG.  154. 


CHROMATIC  VARIATION  OF  JTHE  I,ATERAI. 
MAGNIFICATION.  M'Q',  M'Q'  are  the  two 
coloured  images  of  an  object  MQ  (not  shown 
in  the  diagram).  M"Q"  is  the  position  of 
the  focussing  plane  ad  justed  so  that  the  two 
coloured  images  are  projected  on  it  from  the 
points  M'  and  M'  in  a  piece  M"Q"  which  is 
the  same  for  both  images. 


M'M"      M"Q' 


and  hence,  if 


M'M'      M'Q'  ' 

y_M^'      v 
'  MQ1 


M'M" 
M'M' 


M"Q" 


MQ 


denote  the  lateral  magnifications  for  the  two  colours,  we  have : 


or 


DY 
Y~ 


M'M" 
M'M'' 

M'M" 
~  M'M' 


M'M' 


M'M' 


—  i. 


If  now  the  segments  M'M",  M'M",  M'M'  and  M'M'  may  be  regarded 
as  small  magnitudes  of  the  first  order,  and  if  we  neglect  magnitudes 

1  A.  KOENIG:  Die  Theorie  der  chromatischen  Aberrationen:  Chapter  VI  of  Die  Theorie 
der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR.  See  pages  342 
and  343. 


§  349.]  Colour-Phenomena.  513 

of  the  second  order  of  smallness,  then 

DY       (         M'M"\f          M"M' 

••»    T 


M'M"      M"M' 

M'M'  +  M'M" 

MW'  fMW^MM'\ 

M'M'^  \  M'M'- M'M" 

M'M' 

M'M'' 


so  that  if  the  abscissae,  with  respect  to  the  vertex  of  the  last  surface 
of  the  centered  system  of  spherical  surfaces,  of  the  points  Mr,  M'  and 
M'  be  denoted  by  u',  u'  and  u',  respectively,  we  have  approximately: 

(417) 

*•  M. 

This  formula  is  given  by  KoENic.1 

ART.  111.     CHROMATIC    VARIATIONS   IN    SPECIAL    CASES. 

349.  Optical  System  consisting  of  a  Single  Lens,  surrounded  on 
both  sides  by  air. 

If  the  optical  system  consists  of  a  single  lens  (m  =  2),  surrounded 
on  both  sides  by  air  (n^  —  n'2  =  i,  n(  =  n),  the  formulae  for  the  chro- 
matic longitudinal  aberration,  as  derived  from  the  difference-equation 
(409),  is  as  follows: 

Du2  =  -  U2  •  u'2  (  r1  -  j-L  J,  -  J2 }  ^;  (418) 


which  vanishes  when  u'2  =  o  or  u'2  =  o  (neither  of  which  cases  need 
be  considered),  and  also  when: 

•r 

HfaJj  =  h2h2J2.  (419) 

Since  (cf.  §  126)   h^  =  alt  h2J2  =  na2  —  a'2,  this  condition  may  also 
be'  written  as  follows  : 


1  A.   KOENIG:  Die   Theorie  der  chromatischen  Aberrationen:    Chapter  VI   of   Die 
Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR.     See 
page  345- 
34 


514  Geometrical  Optics,  Chapter  XIII.  [  §  349 

and,  hence,  we  may  say  that  the  condition  of  the  abolition  of  the  chro- 
matic longitudinal  aberration  with  respect  to  an  object-point  on  the 
axis  of  the  lens  is  that  the  angles  of  incidence  at  the  first  surface  and 
of  emergence  at  the  second  surface  of  a  ray  of  the  first  colour  shall  be 
inversely  proportional  to  the  incidence-heights  of  the  corresponding 
ray  of  the  second  colour. 

Equation  (419)  may  also  be  written  as  follows: 

ttX/!  =  u.2u2J2.  (420) 

Now 

_       i       i  /i         i\./i        i   \ 

/!  =  -—-  =  n[ r  )  =  n\ r  )  : 

1      r,      wt         V*i       *iV          \ri      «i/' 

and  hence: 

Dn 


moreover, 


where  d  denotes  the  thickness  of  the  lens.  Eliminating  Jlt  J2  from 
equation  (420)  by  means  of  the  above  relations,  and,  finally,  solving 
for  r2)  we  obtain: 

(u{  -d)(u[  -d)(n-ri) 
r*  ~  .  «(«;  -  d)  -  n(u(  -  d)  ; 

which  is  a  formula  due  to  KESSLER.1  Thus,  if  we  know  the  radius  rt 
of  the  first  surface  of  the  lens  and  the  lens-thickness  d,  this  formula 
enables  us  to  calculate  the  radius  r2  of  the  second  surface,  so  that 
the  images  of  a  given  axial  radiant  point  Mlt  of  colours  corresponding 
to  the  given  values  of  n  and  n,  will  be  formed  at  one  and  the  same 
point. 

In  case  the  axial  object-point  is  infinitely  distant  (%  =  oo),  so  that 
the  focal  points  of  the  lens  (£',  Er)  for  the  two  colours  are  coincident, 
we  find: 

(nrt  —  (n  —  i)d]  {nr^  —  (n  —  i)d}  ^ 
nnr^  —  (w  —  i)(w  —  i)d 

and  it  is  easy  to  show  that  for  such  a  lens  the  secondary  principal 

1F.  KESSLER:  Ueber  Achromasie:  Zft.  f.  Math.  u.  Phys.,  xxix  (1884),  1-24.  This 
admirable  paper  contains  a  complete  treatment  of  the  conditions  of  the  achromatism  of 
a  thick  lens,  with  a  number  of  elegant  geometrical  constructions  by  the  methods  of  pro- 
jective  geometry. 


§  349.]  Colour  Phenomena.  515 

point  of  the  lens  (Ar)  for  the  colour  corresponding  to  the  value  n 
coincides  with  the  secondary  focal  point  of  the  first  surface  of  the 
lens  (Ej)  for  the  colour  corresponding  to  the  value  n\  and,  similarly, 
that  the  point  A'  (secondary  principal  point  of  the  lens  for  the  colour 
corresponding  to  the  value  n)  coincides  with  the  point  E[  (secondary 
focal  point  of  the  first  surface  of  the  lens  for  the  colour  corresponding 
to  the  value  n). 

In  the  case  of  a  lens  surrounded  by  the  same  medium  on  both  sides, 
the  lateral  magnifications  for  the  two  colours  corresponding  to  the 
values  n  and  n  are  as  follows  : 


whence  we  derive  the  following  difference-formula  : 

DY      u(  nr2  -  (n  -  i)(u\  -  d) 
~Y'-^nr2-(n-i)(u{-d)~ 

In  order,  therefore,  that  the  chromatic  variation  of  the  lateral  magni- 
fication shall  vanish  (DY  =  o),  we  have  the  following  condition: 

(n-n)u\-u(  -  {(n-  i)u[  -  (n  -  i]u(}d 
r*  =  nu't-nu'i 

so  that,  provided  the  position  of  the  axial  object-point  is  assigned, 
and  the  radius  of  the  first  surface  of  the  lens  is  given,  together  with 
the  values  of  n  and  n,  this  formula  (423)  gives  r2  and  d  as  linear  funct- 
ions of  each  other. 

Eliminating  u[  and  u{  by  means  of  the  relations  given  above  just 
after  formula  (420),  we  derive  from  formula  (423)  the  following  equa- 
tion: 

-  r,d  ,      x 

wi  = 


whereby  for  a  given  lens  we  can  determine  the  position  of  the  object- 
plane  which  for  two  given  colours  is  portrayed  by  the  lens  in  images 
of  equal  dimensions. 

If  the  thickness  of  the  lens  is 

nn(rl  -  r2) 
a  =  -  -  -  , 
nn  —  I 

then,  according  to  formula  (424),  we  find  WL  =  oo  ;  so  that  for  such  a 
lens  the  object  must  be  situated  at  infinity.  We  find  also  that  the 


516  Geometrical  Optics,  Chapter  XIII.  [  §  350. 

focal  lengths  are  equal ;  thus : 

f  =  7  =        »n-i         ^    r,r2 

J  J  f  A*  T   \    (*~  -,\  ~       * 


and  it  may  also  be  shown  that  the  primary  principal  point  of  this  lens 
(A)  for  the  first  colour  coincides  with  the  secondary  focal  point  of 
the  first  surface  of  the  lens  (E{)  for  the  second  colour;  and,  similarly, 
that  A  coincides  with  E(. 

350.     Infinitely  Thin  Lens. 

If  the  lens  is  infinitely  thin,  we  may  put: 

h±  =  hl  =  h2  =  h2 

in  formula  (418) ;  and  if  also  we  introduce  here  our  special  notation  for 
the  case  of  an  infinitely  thin  lens  (§  268),  so  that  x=i/u,  x'  =  i/u',  and 
x'=i/u'  denote  the  reciprocals  of  the  intercepts  on  the  axis  of  the  inci- 
dent and  emergent  paraxial  rays  for  the  two  colours ;  and  if  <p=i/f  de- 
notes the  "power"  of  the  lens  for  the  colour  corresponding  to  the  value 
n\  and,  finally,  if  c=i/rl,  c'  =  i  fr2 denote  the  curvatures  of  the  bound- 
ing surfaces  of  the  lens,  then  the  lens-formulae  may  be  written  as 
follows : 

<f>  =  (n  -  i)(c  -  c'},     x'  =  x  +  <p. 

Accordingly,  we  derive  the  following  formula  for  the  chromatic  longi- 
tudinal aberration  of  an  infinitely  thin  lens  : 

,_     Dn        _<P 
where 

-;    •'••'•  •          '-^r     v.:;V  :--'     (426) 

is  the  magnitude  defined  in  §  329.  The  reciprocal  value  i/v  is  some- 
times called  the  dispersor  of  the  lens,  and  the  quotient  <p\v  is  called  the 
dispersive  strength  of  the  lens. 

If  M',  M'  designate  the  positions  of  the  image-points  in  the  two 
colours  corresponding  to  the  axial  object-point  M,  then 

Du'  =  M'M'  =  -  u'ii'  -  ; 
v 

and  hence  (except  in  the  merely  theoretical  case  when  u'  =  u'  =  o) 
it  is  not  possible  to  abolish  the  chromatic  longitudinal  aberration  of 
an  infinitely  thin  lens. 


§  351.]  Colour-Phenomena.  517 

When  the  incident  ray  is  parallel  to  the  axis,  we  have  u'  —  /,  u'  =  /, 
and  in  this  case : 

Du'  =  E'E'  =  -^> 
v 

If  the  lens  is  convergent  (/>o),  and  if,  as  is  assumed  throughout, 
n  >  n,  then  Du'  <  o;  so  that  the  more  refrangible  (blue)  rays  are 
brought  to  a  focus  Ef  nearer  to  the  lens  than  the  focus  Ef  of  the  less 
refrangible  (red)  rays;  which  is  the  case  known  as  Chromatic  Under- 
Correction.  The  opposite  effect,  viz.,  Chromatic  Over -Correction,  is  ex- 
hibited by  an  infinitely  thin  divergent  lens. 

The  chromatic  aberration  of  the  lateral  magnification  of  an  infinitely 
thin  lens  is: 

Du'           u'-u'    <p  N 

DY  =  — r  = -•  (427) 

U  U  V 

351.     Chromatic  Aberration  of  a  System  of  Infinitely  Thin  Lenses. 

The  formula  for  the  chromatic  longitudinal  aberration  of  a  system 
of  infinitely  thin  lenses  with  the  centres  of  their  surfaces  ranged  along 
one  and  the  same  straight  line  may  be  derived  very  easily  from  the 
general  formula  (409).  However,  as  we  use  here  a  special  notation 
corresponding  to  that  employed  above  in  the  case  of  a  single  infinitely 
thin  lens,  and  as  the  subscripts  attached  to  the  symbols  relate  now  to  the 
number  of  the  lens,  and  not  to  the  number  of  the  refracting  surface,  it  is 
more  convenient  to  deduce  the  formula  independently.  Consider, 
therefore,  the  kih  lens  of  the  system,  and  let  Ak  designate  the  position 
of  the  optical  centre  of  this  lens.  Also,  let  Mk_u  M'k  designate 
the  points  where  a  paraxial  ray  (emanating  originally  from  the  axial 
object-point  MJ,  of  colour  corresponding  to  the  value  nk,  crosses  the 
optical  axis  before  and  after  refraction,  respectively,  through  this  lens; 
and  let 

-  =  uh  =  AkMk_lt    -7  =  «;  =  AkMk\ 
xk  ** 

and,  similarly,  for  a  paraxial  ray  of  colour  corresponding  to  the  value 
nkJ  which  emanates  from  the  same  axial  object-point  Mlt  we  shall 
write : 

=•  =  uh  =  AkMk_lt     =;=«!  =  AkMk. 
xk  *k 

Denoting  the  strength  or  "power"  of  the  &th  lens  for  the  two  colours 
by  <pk  and  <pk,  we  derive  easily  the  following  difference-relation: 

Dxk  =  Dxk  +  D<pk  =  Dxk  +      , 


518  Geometrical  Optics,  Chapter  XIII.  [  §  351. 

where 


Dn, 


denotes  the  value  of  v  for  the  substance  of  the  kth  lens.    Moreover, 


where  hk,  Tik  denote  the  incidence-heights  of  the  rays  of  the  two  colours 
at  the  kth  lens.     Hence,  we  obtain  the  following  recurrent  formula: 


h 

n 


whence,  putting  Dxl  =  o,  Dx\  =  <pjvlt  we  deduce  easily  the  following 
equation  for  the  chromatic  longitudinal  aberration  of  a  system  of  m 
infinitely  thin  lenses: 


It  may  be  remarked  that  the  actual  forms  of  the  lenses  which  compose 
the  system  are  entirely  immaterial;  so  that  any  lens  in  the  system 
may  be  replaced  by  an  equivalent  one  of  different  form  but  of  the 
same  dispersive  strength  (§  350),  without  affecting  the  magnitude  of 
the  chromatic  longitudinal  aberration. 

If  the  distance  between  each  pair  of  lenses  of  the  system  is  negligible, 
that  is,  if  we  have  a  system  of  m  infinitely  thin  lenses  in  contact,  the 
magnitudes  denoted  by  the  letters  h  will  all  be  equal,  and  in  this  case 
we  shall  have: 


* 

k=l 


!>*:=£  *;  (429) 


so  that  for  such  a  system  the  chromatic  longitudinal  aberration  is 
independent  of  the  position  of  the  axial  object-point;  and,  moreover, 
the  order  in  which  the  lenses  are  placed  does  not  matter. 
A  single  infinitely  thin  lens  of  strength 


k=m 
k=l 


§  352.]  Colour-Phenomena.  519 

and  made  of  material  whose  v- value  is: 


(431) 


will  be  equivalent  to  the  system  of  m  thin  lenses  in  contact,  in  respect 
both  to  the  refraction  and  the  dispersion  of  paraxial  rays.  Thus,  for 
a  given  value  of  <p  and  for  a  given  value  of  vk,  we  may  vary  the  strength 
<pk  of  the  &th  lens,  so  that  v  has  any  arbitrary  value  whatever.  This 
fact  is  of  immense  importance  to  the  optician;  for  although  he  has  at 
his  disposal  only  a  limited  series  of  optical  glasses  with  values  of  v 
ranging  from,  say,  v  =  20  to  v  =  70,  yet  in  case  he  needs  for  a  certain 
lens  a  certain  v-value  that  does  not  belong  to  any  actual  kind  of  glass, 
he  has  merely  to  substitute  for  this  lens  a  suitable  combination  of  two 
or  more  lenses.1 

If  the  system  of  thin  lenses  in  contact  is  achromatic,  we  have: 


in  which  case  the  rvalue  of  the  combination  is  equal  to  infinity. 

352.  In  particular,  let  us  consider,  first,  a  system  consisting  of  Two 
Infinitely  Thin  Lenses  in  Contact  (m  =  2).  The  condition  of  the 
abolition  of  the  chromatic  aberration  with  respect  to  the  place  of  the 
image,  as  derived  from  formula  (429),  is  as  follows: 

^i  ,   V>2 
—  —  =  o. 

"l  "2 

If  the  differences  of  the  curvatures  of  the  two  surfaces  of  the  lenses 
be  denoted  by  Ci  and  C2,  that  is,  if 

Cx=  cl  —  c(,     C2  =  c2  -  c'2,  (433) 

then,  since 

<Pi  =  (n\  ~  i)  Q*     <Pz  =  (w2  -  0  C2t 


the  condition  above  may  be  expressed  in  the  following  form  also: 

C;  •£>»!  +  C2-Dn2  =  o.  (434) 

1  See  A.  KOENIG:  Die  Theorie  der  chromatischen  Aberrationen,  Chapter  VI  of  Die 
Theorie  der  optischen  Inslrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR.  See 
page  349. 


520  Geometrical  Optics,  Chapter  XIII.  [  §  353. 

Moreover,  if 

9    =    <Pl  +   ¥?2 

denotes  the  strength  of  the  combination  of  lenses,  we  find  that,  in 
order  to  fulfil  the  condition  of  achromatism,  the  strengths  of  the  two 
lenses  must  be  as  follows: 


also  we  obtain: 


According  to  these  results,  a  system  of  two  infinitely  thin  lenses 
in  contact  can  be  achromatic  only  in  case  the  p- values  of  the  two  lenses 
are  different;  that  is,  the  two  lenses  must  be  made  of  different  kinds 
of  glass.  Moreover,  the  focal  lengths  of  the  lenses  must  have  opposite 
signs,  and  must  be  inversely  proportional  to  the  ^-values.  The  focal 
length  of  the  combination  has  the  same  sign  as  that  of  the  lens  with 
the  greater  v- value ;  thus,  the  strength  (ip)  of  the  achromatic  combina- 
tion will  be  positive  when  the  strength  of  the  positive  lens  exceeds 
that  of  the  negative  lens.  For  a  prescribed  value  of  <p,  the  strengths 
of  the  individual  lenses  are  smaller  in  proportion  as  the  rvalues  are 
smaller,  and  also  in  proportion  as  the  difference  of  the  ^-values  is 
greater.  By  selecting  two  kinds  of  glass  with  as  great  difference  of 
rvalues  as  possible,  we  can  reduce  the  differences  of  the  curvatures 
of  the  two  surfaces  of  the  lenses. 

353.  System  of  Two  Infinitely  Thin  Lenses  Separated  by  a  Finite 
Interval  (d). 

According  to  formula  (428),  the  chromatic  longitudinal  aberration 
of  a  system  of  two  infinitely  thin  lenses  is  as  follows: 


which,  since 
b 


(where  d  ==•  A^2  denotes  the  distance  between  the  lenses),  may  be 
written  in  the  following  form  : 


7  +  7 


§  353.]  Colour-Phenomena.  521 

It  can  be  easily  shown  that,  except  in  the  case  when  the  second 
lens  is  so  placed  as  to  separate  the  two  coloured  images  M  (  ,  M  (  formed 
by  the  first  lens,  the  strengths  of  the  two  lenses  must  have  opposite 
signs  in  order  that  Dx2  shall  vanish.  If  we  may  assume  that  the 
variations  ZVi=s^i/I'i  anc^  D<?2  —  ^2/^2  are  so  small  that  their  product 
D<p±-D(p2  may  be  neglected,  the  condition  that  the  system  of  two  thin 
lenses  shall  have  the  same  focal  point  for  the  two  colours  is  found 
by  putting  xl  =  o  in  the  equation  Dx'2  =  o  ;  which  condition  is,  there- 
fore, as  follows: 


If  the  two  lenses  are  made  of  the  same  kind  of  glass  (^  =  j>2),  the 
condition  that  Dx'2  =  o  becomes: 

HjiM  +  hji2<p2  =  o; 

which  is  analogous  to  the  condition,  expressed  in  formula  (419),  for 

the  abolition  of  the  chromatic  longitudinal  aberration  of  a  thick  lens. 

The  angular  magnification  (Z)  of  a  system  of  two  infinitely  thin 

separated  lenses  is: 

/       / 
_  xl    x2 

•:*i'*99 

whence,  since 

/ 

x(  =  xl  +  <ft,     x2  =  x2  4-  <P2>    *2  =  r    v^S  . 

1    ~~  JL-j  *  Uf 

we  obtain: 


Accordingly,  the  formula  for  the  chromatic  magnification-difference 
(DZ)  is  as  follows: 


Assuming  here  also  that  the  variations  D^  and  D<p2  are  so  small  that 
we  may  neglect  their  product,  we  can  write  the  condition  of  the  aboli- 
tion of  the  chromatic  magnification-difference  as  follows: 


which,  in  general,  will  not  be  independent  of  the  position  of  the  object. 
If  this  condition  is  fulfilled,  not  only  will  the  two  coloured  rays  emerge 


522  Geometrical  Optics,  Chapter  XIII.  [  §  353. 

from  the  system  in  parallel  directions,  but  the  two  coloured  images 
will  be  of  equal  size;  since  DY  vanishes  along  with  DZ  (see  §  347). 
If,  therefore,  the  two  lenses  are  made  of  the  same  material  fa  =  i>2), 
they  must  be  separated  by  the  distance 


if  the  two  coloured  images  of  the  object  are  to  be  equal  in  size.  It 
will  be  observed  that  here  d  is  independent  of  the  value  of  v,  and 
herein  consists  the  great  advantage  of  employing  a  combination  of 
lenses  of  the  same  kind  of  glass;  for  in  such  a  case  if  two  coloured 
images  are  made  of  equal  size,  all  the  coloured  images  will  be  of  equal 
size. 

When  the  object  is  infinitely  distant  (xl  —  o),  the  formula  above 
will  be  simplified  as  follows: 


that  is,  the  distance  between  the  two  lenses  must  be  equal  to  half  the 
sum  of  their  focal  lengths. 

The  most  important  application  of  this  last  result  is  in  the  construct- 
ion of  the  oculars  of  telescopes.1  If  the  actual  paths  of  the  rays  through 
a  telescope  are  supposed  to  be  reversed  in  direction,  then  the  ocular 
will  produce  in  the  focal  plane  of  the  object-glass  a  real  image  of  an 
infinitely  distant  object,  and  the  lens  farthest  from  the  object-glass 
will  be  the  first  lens  of  the  system  of  two  lenses  which  form  the  ocular. 
The  ocular  called  after  HUYGENS  is  composed  of  two  lenses  as  follows: 


and  the  construction-data  of  the  RAMSDEN-ocular  are  as  follows: 


so  that  both  of  these  types  satisfy  the  condition  expressed  in  the  last 
formula. 

It  may  be  added,  in  conclusion,  that  it  is  not  possible  to  have  com- 
plete achromatism  of  a  system  of  two  lenses  separated  by  a  finite 
interval. 

1  Concerning  this  matter,  see  E.  VON  HOEGH:  Die  achromatische  Wirkung  der  HUY- 
GHENs'schen  Okulare.  Central-  Zeitung  /.  Opt.  u.  Mech.,  vii  (1886),  37,  38. 


§354.]  Colour-Phenomena.  523 

ART.    112.     THE   SECONDARY   SPECTRUM. 

354.  In  consequence  of  the  so-called  "irrationality  of  the  dispers- 
ion" (§  329),  it  is  evident  that  even  when  an  optical  system  has  been 
designed  so  as  to  be  achromatic  with  respect  to  one  pair  of  colours, 
it  will,  in  general,  not  be  achromatic  for  all  colours.  If,  for  example, 
it  is  contrived  so  that  the  red  and  blue  rays  are  again  united  in  the 
image,  there  will  still  be,  perhaps,  a  slight  dispersion  of  the  yellow 
and  green  rays;  that  is,  an  uncorrected  residual  colour-error,  or,  as 
BLAIR  termed  it,  a  "secondary  spectrum".1  The  residual  chromatic 
longitudinal  aberration  of  a  system  of  thin  lenses  which  is  achromatic 
with  respect  to  two  principal  colours  has  been  thoroughly  investigated 
by  KOENIG  ;  2  whose  methods  will  be  used  in  the  brief  and  elementary 
treatment  of  this  matter  that  is  given  here,  wherein  we  consider  only 
the  secondary  spectrum  of  a  system  of  thin  lenses  in  contact. 

If  n,  n  denote  the  indices  of  refraction  of  an  optical  medium  for 
the  two  principal  colours  with  respect  to  which  the  optical  system  is 
assumed  to  be  achromatic,  the  difference  n  —  n  =  Dn  is  called  the 
fundamental  dispersion  of  the  medium;  and  if  n  denotes  the  index  of 
refraction  of  the  same  medium  for  a  third  colour,  the  difference 
&n  =  n  —  n  is  called  the  partial  dispersion;  and  the  ratio 

Qn      n  —  n 

.      Hr-jrrs-*  '    (438) 

is  called  the  relative  partial  dispersion  of  the  medium  (see  §  329).  In 
general,  the  relative  partial  dispersion  ft  of  one  medium  will  be  dif- 
ferent from  the  relative  partial  dispersion  ft  of  another  medium. 
The  ratio 

n  —  I      v 


will  be  the  p-value  of  the  medium  for  the  interval  from  the  first  to 
the  third  colour. 

If  the  chromatic  longitudinal  aberration  with  respect  to  the  two 
principal  colours  has  been  abolished,  the  paraxial  image-rays  corre- 
sponding to  these  two  colours  which  emanate  originally  from  the  axial 

1  See  S.  CZAPSKI:  Mittheilungen  ueber  das  glastechnische  Laboratorium  in  Jena  und 
die  von  ihm  hergestellten  neuen  optischen  Glaeser:   Zft.  f.  Inshumentenkun.de,  vi  (1886), 
293-299,  335-348.     Also,  see  S.  CZAPSKI:   Theorie  der  optischen  Instrumente  nach  ABBE 
(Breslau,  1893),  pages  128-132. 

2  A.  KOENIG:  Die  Theorie  der  chromatischen  Aberrattonen  :  Chapter  VI  of  Die  Theorie 
der  optischen  Instrumente,  Bd.  I   (Berlin,  1904),  edited  by  M.  VON  ROHR.     See  pages 
357-366. 


524  Geometrical  Optics,  Chapter  XIII.  [  §  354. 

object-point  M,  will  cross  the  optical  axis  at  one  and  the  same  point 
M ' ;  and  if  2ft'  designates  the  point  where  the  paraxial  image-rays  cor- 
responding to  the  third  colour  cross  the  axis,  then,  following  KOENIG, 
let  us  agree  to  call  the  line-segment 

M'm*  =  u'  -  u' 

(where  «',  u'  denote  the  distances  from  the  system  of  thin  lenses  in 
contact  of  the  points  M' ,  9ft',  respectively)  the  secondary  spectrum  with 
respect  to  this  third  colour. 

For  example,  consider  an  achromatic  combination  of  m  thin  lenses 
in  contact,  for  which  the  condition 

k=m 


is  satisfied.  According  to  formula  (429),  the  magnitude  of  the  second- 
ary spectrum  will  be : 

wherein  we  have  put  um  =  u'm'\\m\  and,  accordingly,  the  condition 
of  the  abolition  of  the  secondary  spectrum  of  an  achromatic  combina- 
tion of  m  thin  lenses  in  contact  is  as  follows : 

k=m        o 

£  ~v~"  =  o .  (440) 

If  the  v- values  are  all  equal,  this  equation  is  equivalent  to  the  condi- 
tion of  achromatism  with  respect  to  the  pair  of  principal  colours. 
Evidently,  a  system  composed  of  two  thin  lenses  in  contact,  made  of 
different  kinds  of  glass,  cannot  be  achromatic  with  respect  to  three 
different  colours.  In  the  case  of  a  system  of  three  thin  lenses  in  con- 
tact, the  conditions  to  be  fulfilled  for  the  abolition  of  the  secondary 
spectrum  are  as  follows: 

Z  <pk  —  <P>    Z  ~  =o,    Z  ~ — *  =  o« 

*=i  *=i  Vk  &=]      vk 


The  magnitude  of  the  secondary  spectrum  of  an  achromatic  combi- 
nation of  two  thin  lenses  in  contact  is  as  follows: 


',' 


§  355.]  Colour  Phenomena.  525 

whence  we  see  that  the  smaller  the  difference  of  the  relative  partial 
dispersions  of  the  two  kinds  of  glass  and  the  greater  the  difference  of 
their  rvalues,  the  less  will  be  the  magnitude  of  the  secondary  spectrum. 
A  number  of  pairs  of  glasses  fulfilling  these  requirements  will  be  found 
listed  in  the  catalogue  of  the  "glastechnische  Laboratorium"  in  Jena. 

355.  The  character  and  extent  of  the  secondary  spectrum  of  an 
achromatic  combination  of  lenses  will  evidently  depend  on  the  choice 
of  the  two  principal  colours  with  respect  to  which  the  conditions  of 
achromatism  are  satisfied.  A  chief  consideration  in  the  determination 
of  the  two  colours  that  are  to  be  united  will  be  the  mode  of  using  the 
instrument.  Thus,  if  it  is  designed  to  be  an  optical  instrument  in  the 
literal  sense  of  that  term,  we  shall  be  concerned  primarily  with  the 
physiological  actions  of  the  rays  on  the  retina  of  the  eye;  whereas 
in  the  case,  for  example,  of  a  photographic  objective,  in  which  the 
rays  are  to  be  focussed  on  a  sensitive  plate,  achromatism  with  respect 
to  the  so-called  actinic  rays  will  be  extremely  desirable. 

The  rays  that  are  most  effective  in  their  actions  on  the  retina  of 
the  eye  are  comprised  between  the  FRAUNHOFER  lines  C  and  F,  with 
a  distinct  maximum  of  brightness  in  the  region  between  the  lines  D 
and  E.  If,  therefore,  the  instrument  is  intended  to  be  used  by  the 
eye,  it  is  usual  to  design  it  so  as  to  be  achromatic  with  respect  to  the 
colours  corresponding  to  C  and  F.  Assuming  that  the  system  is  a 
convergent  combination  of  two  thin  lenses  in  contact,  we  shall  find 
then  that  the  focal  points  corresponding  to  the  colours  between  C  and 
F  will  lie  nearer  to  the  lens-system,  and  the  focal  points  corresponding 
to  the  other  colours  will  lie  farther  from  it,  than  the  common  focal 
point  of  the  two  principal  colours.  Moreover,  the  secondary  spectrum 
will  be  approximately  least  for  some  colour  very  nearly  corresponding 
to  the  ZMine,  which  is  a  very  favourable  circumstance,  since  this  is  the 
brightest  region  of  the  spectrum  for  visual  purposes. 

For  the  purposes  of  astrophotography,  it  is  found  best  to  obtain 
as  great  a  concentration  as  possible  of  the  actinic  rays,  especially  as 
here  the  object  will  usually  be  of  relatively  feeble  light-intensity. 
Moreover,  since  the  celestial  objects  are  infinitely  distant,  the  focus- 
sing of  the  instrument  may  be  done  once  for  all,  so  that  the  eye  does 
not  have  to  judge  of  the  perfection  of  this  adjustment,  and  conse- 
quently we  may  disregard  the  visual  rays  here  entirely.  Such  an 
instrument  will  be  designed,  therefore,  to  unite  the  rays  corresponding 
(say)  to  the  FRAUNHOFER  line  F  and  the  violet  line  in  the  spectrum  of 
mercury.  The  secondary  spectrum  with  respect  to  the  longer  wave- 
lengths will  be  very  extensive,  but  in  this  case  this  will  not  matter. 


526  Geometrical  Optics,  Chapter  XIII.  [  §  357. 

In  the  case  of  the  ordinary  photographic  objective,  however,  the 
conditions  are  different,  and  here  it  is  found  necessary  to  effect  a  com- 
promise between  the  visual  and  the  actinic  rays;  because  the  image 
has  to  be  focussed  first  on  the  ground-glass  plate  by  the  eye,  and 
afterwards  the  rays  have  to  be  received  on  the  sensitive  plate.  Under 
these  circumstances  the  usual  procedure  is  to  make  the  system  achro- 
matic with  respect  to  the  colours  corresponding  to  the  lines  D  and  G'.1 

ART.    113.     CHROMATIC   VARIATIONS    OF   THE    SPHERICAL    ABERRATIONS. 

356.  If  the  aperture  of  the  optical  system  is  not  infinitely  narrow 
(as  has  been  assumed  throughout  in  the  preceding  investigation  of 
the  Chromatic  Aberrations),  not  only  the  paraxial  rays  but  also  the 
rays  that  meet  the  objective  at  points  lying  outside  this  small  central 
zone  will  be  concerned  in  the  formation  of  the  image;  so  that  even  if 
the  chromatic  aberrations  of  the  central  rays  and  the  spherical  aber- 
rations of  the  outer  rays  could  all  be  abolished,  the  image  would  still 
not  be  perfect  on  account  of  faults  due  to  the  colour-dispersion  of 
the  latter  rays.     These  faults,  it  is  true,  will  not  be  very  objectionable 
so  long  as  the  aperture  of  the  system  is  relatively  small;  but  with 
systems  of  large  aperture  the  definition  of  the  image  may  be  seriously 
impaired,  and  if  we  wish  to  obtain  an  image  that  is  entirely  free  from 
colour-faults,  the  chromatic  aberration  must  be  abolished  for  each  zone 
of  the  objective. 

It  would  be  a  waste  of  time  to  develop  here  difference-formulae 
for  the  chromatic  variations  of  SEIDEL'S  approximate  expressions  of 
the  five  spherical  aberrations;  because  it  is  only  when  the  rays  make 
considerable  angles  with  the  optical  axis  that  the  colour-faults  under 
consideration  assume  serious  importance.  Indeed,  it  will  only  be  nec- 
essary to  consider  briefly  the  two  of  these  faults  that  are  the  most 
troublesome  in  the  ordinary  case  of  optical  systems  of  large  aperture, 
viz.,  the  chromatic  variations  (i)  of  the  Longitudinal  or  so-called 
"Spherical"  Aberration  and  (2)  of  the  Aplanatism  or  Sine-Ratio. 

357.  Let  us  begin  with  an  investigation  of  the  colour-variation 
from  zone  to  zone  of  the  place  where  the  rays  cross  the  optical  axis, 
called  by  ABBE2  "the  chromatic  difference  of  the  spherical  aberration"; 
which,  as  KRUESSS  remarks,  may  also  be  called  with  equal  right  "the 

1  This  whole  matter  is  thoroughly  explained  and  discussed  in  M.  VON  ROHR'S  Theorie 
und  Geschichte  des  photographischen  Objektivs  (Berlin,  1899),  pages  60-64.  See  also  A. 
KOENIG:  Die  Theorie  der  chromatischen  Aberrationen,  Chapter  VI  of  Die  Theorie  der 
oplischen  Inslrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR;  pages  366-369. 

1  E.  ABBE:  On  New  Methods  for  Improving  Spherical  Correction,  applied  to  the  Con- 
struction of  Wide-Angled  Object-glasses:  Roy.  Mic.  Soc.  Journal,  (2)  2,  (1879),  812-824. 
See  also:  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904),  196-212. 

8H.  KRUESS:  Die  Farben-Correction  der  Fernrohr-Objective  von  GAUSS  und  von 
FRAUNHOFER:  Zft.  f.  Instr.,  viii  (1888),  7-13,  53-63  and  83-95. 


§  357.] 


Colour- Phenomena. 


527 


spherical  difference  of  the  chromatic  aberration",  whereby  this  colour- 
fault  is  regarded  as  due  to  the  variation  of  the  chromatic  longitudinal 
aberration  (§  346)  from  zone  to  zone. 

The  adjoining  diagrams  (Figs.  155  and  156),  similar  to  those  given 
by  LuMMER1  in  his  treatment  of  this  subject,  will  help  to  make  the 
matter  clear.  In  both  figures  the  red  rays  and  the  blue  rays  repre- 
senting the  light  of  the  longer  wave-lengths  and  the  shorter  wave- 
lengths, respectively,  are  shown  on  opposite  sides  of  the  optical  axis; 
thus,  above  the  axis  the  two  rays  selected  are  a  red  paraxial  ray  and 
a  red  edge-ray;  whereas  below  the  axis  the  two  corresponding  rays 
are  blue.  For  some  colour, 
say,  yellow  (as  being  op- 
tically the  most  intensive), 
intermediate  between  red 
and  blue,  the  optical  sys- 
tem in  both  cases  is  sup- 
posed to  be  spherically 
corrected,  so  that  the  edge-  FIG. 

rays   corresponding    to  this          OPTICAL  SYSTEM  m  WHICH  THE  CHROMATIC  I,ONGI- 
.  .         TUDINAL  ABERRATION  OF  THE  CENTRAL  RED  AND  BLUE 

mean  colour  cross  the  axis     RAYS  is  ABOLISHED. 
at  the  same  point  as  the 

central  rays  of  this  colour.  In  both  cases  also  there  is  spherical 
under-correction  of  the  red  rays  and  spherical  over-correction  of  the 
blue  rays.  In  Fig.  155,  however,  the  chromatic  longitudinal  aberra- 
tion of  the  central  red  and 
blue  rays  is  abolished, 
whereas  in  Fig.  156  the 
chromatic  longitudinal 
aberration  of  the  red  and 
blue  edge-rays  is  abol- 
ished. In  both  illustra- 
tions there  is  a  residual 
chromatic  aberration, 
which  in  the  case  of  sys- 
tems of  relatively  large 

aperture  may  be  more  injurious  than  the  so-called  "secondary  spec- 
trum" (Art.  112)  due  to  the  disproportionality  of  the  dispersion-ratios 
for  the  different  parts  of  the  spectrum. 


FIG.  156. 

OPTICAL  SYSTEM  IN  WHICH  THE  CHROMATIC  I^ONGITU- 
DINAL  ABERRATION  OF  THE  RED  AND  BLUE  EDGE-RAYS 
IS  ABOLISHED. 


1  O.  LUMMER:  See  MUELLER-POUILLET'S  Lehrbuch  der  Physik  und  Meteorologie,  Bd. 
II,  zehnte  Auflage  (Braunschweig,  1909),  Art.  169. 


528  Geometrical  Optics,  Chapter  XIII.  [  §  357. 

As  far  back  as  1817,  GAUSS1  "suggested  a  plan  for  getting  rid  of 
the  residual  aberration  in  binary  lenses  made  of  ordinary  crown  and 
flint,  for  telescopic  use".2  Thus,  if  the  optical  system  is  so  contrived 
that  there  is  spherical  correction  of  the  extreme  red  rays  and  chro- 
matic correction  of  the  red  and  blue  paraxial  rays,  GAUSS'S  Condition, 
as  it  is  called,3  requires  that  we  shall  have  also  the  extreme  blue  rays 
crossing  the  optical  axis  at  the  same  point  as  the  extreme  red  and  the 
paraxial  red  and  blue  rays.4  As  a  rule,  the  fulfilment  of  GAUSS'S 
Condition  will  necessitate  lenses  of  very  deep  curvatures. 

Since  it  is  not  possible  to  have  perfect  chromatic  correction  for  the 
entire  aperture  of  the  objective,  the  question  arises  for  what  zone  is 
it  best  to  abolish  the  chromatic  difference  of  the  spherical  aberration, 
so  that  the  defects  in  the  image  on  this  score  may  be  diminished  as 
much  as  possible.  As  KoENiG5  remarks,  this  question  cannot  be  de- 
cided by  geometrical  optics  alone;  but  the  matter  has  been  investi- 
gated from  this  point  of  view  by  KERBERS  who  suggested  that  it  was 
natural  to  endeavor  to  obtain  chromatic  correction  for  that  place  on 
the  optical  axis  which  is  determined  by  the  position  of  the  transversal 
plane  of  least  cross-section  of  the  bundle  of  image-rays  corresponding 
to  the  principal  colour.  Thus,  if  H  denotes  the  incidence-height 
at  the  last  spherical  surface  of  the  extreme  outside  rays  of  a  bundle  of 
image-rays  of  a  given  colour,  emanating  originally  from  an  axial  object- 
point,  and  if  the  point  where  the  optical  axis  meets  the  transversal 
plane  of  least  cross-section  of  this  bundle  of  image-rays  is  designated 
by  N',  it  may  be  shown,  by  simple  deductions  from  the  results  of 
§  351, that 

AN' =  u' +  *^.B*; 

4  u' 

where  u'  denotes  the  abscissa,  with  respect  to  the  vertex  A  of  the  last 
spherical  surface,  of  the  point  M'  where  the  paraxial  rays  corresponding 

1  C.  F.  GAUSS:  Ueber  die  achromatischen  Doppelobjective  besonders  in  Ruecksicht 
der  vollkommenern  Aufhebung  der  Farbenzerstreuung:   Zft.  f.  Astron.  M.  verwandte  Wis- 
senschaften,  IV  (1817),  343-351.      See  also:    GAUSS'S  Werke,  Bd.  V   (zweiter  Abdruck, 
Goettingen,  1877),  504-508. 

2  See  E.  ABBE:  On  New  Methods  for  Improving  Spherical  Correction,  etc.:  Roy.  Mic. 
Soc.  Journ.,  (2)  2  (1879),  p.  814. 

8  H.  KRUESS:  Die  Farben- Correction  der  Fernrohr-Objective  von  GAUSS  und  von 
FRAUNHOFER:  Zft.  f.  Instr.,  viii  (1888),  7-13,  53-63  and  83-95. 

4  See  also  S.  CZAPSKI:  Theotie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893), 
P-  134- 

5  A.  KOENIG:  Die  Theorie  der  chromatischen  Aberrationen,   Chapter  VI  of  M.  VON 
ROHR'S  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  p.  370. 

6  A.    KERBER:  Ueber   die   chromatische   Korrektur   von   Doppelobjektiven:   Central- 
Zeitungf.  Opt.  u.  Mech.,  vii  (1886),  157-158. 


§  357.]  Colour-Phenomena.  529 

to  this  colour  cross  the  optical  axis  (uf  =  A  M'}  ;  and  where  a'  is  the 
characteristic  aberration-co-efftcient  employed  in  the  series-develop- 
ment in  formula  (273).  If  here  we  use  the  symbol  h  to  denote  the 
incidence-height  of  the  cone  of  rays  of  this  bundle  of  image-rays  that 
meet  the  axis  at  the  point  N',  we  have  also: 

AN'  -  »'  +  ^  •  h2; 


u 


and,  hence,  by  equating  these  two  expressions  for  the  abscissa  AN', 
we  find: 


as  given  by  KERBER;  who  concludes,  according  to  this  process  of 
reasoning,  that  the  chromatic  correction  should  be  made  for  the  zone 
whose  height  above  the  axis  is  h  =  0.866-  H. 

A  comprehensive  view  of  the  performance  of  a  given  optical  system 
with  respect  to  the  chromatic  variations  of  the  spherical  aberrations 
for  rays  of  different  colours  and  of  different  incidence-heights  can  be 
obtained  by  means  of  the  so-called  isoplethic  curves  employed  by  M. 
VON  RoHR.1  The  wave-lengths  of  the  light  (expressed  in  ju/i)  are  laid 
off  along  the  axis  of  abscissae,  whereas  the  incidence-heights  (in  mm.) 
are  represented  along  the  other  of  the  two  rectangular  axes  of  the 
diagram;  so  that  to  each  point  in  the  plane  of  the  figure  there  corre- 
sponds a  certain  ray  of  a  definite  colour  and  of  a  definite  incidence- 
height.  In  the  object-space  of  the  optical  system  the  rays  are  all 
assumed  to  be  parallel  to  the  optical  axis.  If  M'  designates  the  point 
where  a  paraxial  image-ray  of  mean  refrangibility,  corresponding,  say, 
to  the  FRAUNHOFER  D-line,  crosses  the  optical  axis,  and  if  U  desig- 
nates the  point  where  a  ray  of  some  other  colour,  say  X,  and  of  finite 
incidence-height  In,  crosses  the  optical  axis,  we  calculate  (in  thousandths 
of  a  millimetre)  the  length  M'L'  \  and  if  the  point  (X,  h)  in  the  diagram 
is  designated  by  P,  we  ascribe  to  this  point  P  the  number  corresponding 
to  the  numerical  value  of  M'lJ  '.  The  curve  drawn  through  all  points 
P  which  have  the  same  numerical  value  will  be  one  of  the  system  of 
isoplethic  curves  of  the  optical  system.  M.  VON  ROHR  gives  diagrams 
showing  the  system  of  isoplethic  curves  from  the  value  M'L'  =  +0.050 
mm.  to  the  value  M'L'  =  —  0.050  mm.  for  a  PETZVAL  portrait-object- 
ive and  for  the  so-called  "Planar"  type  of  photographic  objective  of 
P.  RUDOLPH. 

1  M.  VON  ROHR:  Theorie  und  Geschichte  des  photographischen  Objektivs  (Berlin,  1899), 
65-68. 
35 


530  Geometrical  Optics,  Chapter  XIII.  [  §  358. 

The  errors  due  to  the  chromatic  variations  of  spherical  aberration 
are  especially  objectionable  in  the  case  of  microscope-objectives  on 
account  of  their  large  apertures;  but  we  have  not  space  to  describe 
here  the  ingenious  and  successful  methods  that  have  been  devised  by 
ABBEI  to  overcome  this  defect  in  such  optical  systems. 

358.  If  the  wide-angle  optical  system  is  to  produce  a  good  image 
not  merely  of  an  axial  object-point,  but  also  of  adjacent  points  not 
on  the  axis,  for  example,  of  a  surface-element  at  right  angles  to  the 
axis,  in  addition  to  the  abolition  of  the  spherical  aberration,  the  Sine- 
Condition,  as  expressed  by  formula  (300),  must  also  be  fulfilled,  viz.: 

sing  _  n^ 
sin  0'  ~  n 

Obviously,  it  will  be  meaningless  to  investigate  the  chromatic  variation 
of  the  sine-ratio  unless  both  the  spherical  aberration  along  the  axis  and 
the  chromatic  differences  thereof  have  been  abolished.  Assuming, 
therefore,  that  these  prerequisite  conditions  have  been  satisfied,  the 
condition  of  Aplanatism  for  two  adjacent  colours  of  the  spectrum,  cor- 
responding to  the  wave-lengths  X  and  \-\-d\,  requires  not  only  that  the 
above  equation  for  X  shall  be  true,  but  that  the  equation  obtained 
therefrom  by  variation  with  respect  to  X  shall  also  be  satisfied,  viz.: 

d6'          dO        (Mdn          dY 


_ 
tan  0'  "  tan  0       n'  ~  '  n  Y  ' 

which  may  be  written  in  the  form  given  by  CzAPSKi2  as  follows: 

dn\  dY 


An  optical  system  which  is  without  secondary  spectrum  and  aplana- 
tic  for  two  colours  has  been  called  by  ABBES  an  "apochromatic" 
system. 

Even  when  the  last  equation  is  satisfied,  there  will  still  be  an  error 
due  to  the  difference  in  the  magnification  of  the  two  coloured  images 
for  each  zone  of  the  objective;  and  if  this  is  to  be  abolished  also,  we 
must  put  dY  =  o;  and,  hence  (supposing  also  that  the  object  itself 

1  E.  ABBE:  On  New  Methods  for  Improving  Spherical  Correction,  applied  to  the  Con- 
struction of  Wide-angled  Object-glasses:  Roy.  Mic.  Soc.  Jour.,  (2)  2  (1879),  812-824. 
See  also:  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904),  196-212. 

1  S.  CZAPSKI:  Die  Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  p.  134. 

3  E.  ABBE:  Ueber  neue  Mikroskope:  SUz.-Ber.  Jen.  Ges.  Med.  u.  Natw.,  1886,  107-128; 
see  p.  in.  Also:  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904),  p.  454. 


§359.] 


Colour-Phenomena. 


531 


is  free  from  aberrations,  so  that  dQ  =  o)  we  obtain: 


def 
tan0' 


dn 
n 


n 


359.  If  v,  0  denote  the  ray-co-ordinates  of  a  ray  of  wave-length  X 
before  refraction  at  a  given  spherical  surface  of  radius  r,  and  if  a,  a' 
denote  the  angles  of  incidence  and  refraction,  respectively,  we  have 
the  following  system  of  equations  for  determining  the  corresponding 
ray-co-ordinates  (v'}  6')  of  the  refracted  ray  (§  211): 


v  —  r  . 
sin  a  = sin  0, 


a  —  a, 


.      ,       n 
sin  a   =  — : 


vr  —  r  =  — 


r  •  sin  a' 
sin0' 


whence  for  an  adjacent  ray  of  wave-length  X  -f-  d\  we  derive  immedi- 
ately a  series  of  differential  formulae  as  follows: 


da 

dQ 

dv 

tana 
da' 

tan0 
da 

v-r' 
(dn'      dn\ 

tana' 

dd' 
dv' 

tana 
=  dO  -\-da 
da' 

\-n'~  *;• 

'-<*a, 
</0' 

v'  -r 

tan  a' 

tan  0'  ' 

(441) 


so  that  if  we  know  the  values  of  dv,  dQ  before  refraction  at  a  given 
spherical  surface,  we  can  find  the  corresponding  variations  dv',  dQ' 
after  refraction. 


CHAPTER    XIV. 

THE  APERTURE  AND  THE  FIELD  OF  VIEW.     BRIGHTNESS  OF  OPTICAL 

IMAGES. 

ART.    114.     THE   PUPILS. 

360.    Effect  of  Stops. 

In  the  ideal  case  of  perfect  collinear  correspondence  between  Object- 
Space  and  Image-Space  we  found  that  it  was  sufficient  to  know  the 
constants  of  the  optical  system,  for  example,  the  magnitudes  of  the 
focal  lengths  and  the  positions  of  the  focal  points,  in  order  to  construct 
the  image  of  a  given  object;  but  the  rays  that  are  employed  in  such 
graphical  constructions  are  seldom,  if  ever,  the  actual  rays  that  are 
utilized  by  the  optical  instrument  in  the  formation  of  an  image.  From 
a  purely  geometrical  point  of  view,  one  pair  of  rays  was  as  good  as 
another  in  locating  the  position  of  an  image-point,  and  we  were  not  at 
all  concerned  to  inquire  whether  the  rays  so  employed  were  really 
operative  or  not.  But  when  it  was  attempted  to  realize  an  optical 
image  by  the  aid  of  a  so-called  optical  instrument  consisting  of  a 
centered  system  of  spherical  refracting  or  reflecting  surfaces,  we  en- 
countered, first  of  all,  the  spherical  aberrations;  and  at  once  the  quest- 
ion as  to  the  actual  rays  that  are  concerned  in  the  phenomena  assumed 
a  place  of  fundamental  importance,  so  that  not  only  the  degree  of 
perfection  of  the  image,  but  the  region  of  validity  of  the  imagery,  and 
the  possibility  of  extending  it,  are  found  to  depend  essentially  on  the 
slopes  of  the  rays,  and  on  the  heights  above  or  below  the  optical  axis 
of  the  points  where  they  meet  the  spherical  surfaces,  and  on  the  apert- 
ure-angles of  the  ray-bundles,  etc. 

As  a  matter  of  fact,  even  under  ideal  conditions  of  collinear  corre- 
spondence between  two  infinitely  extended  space-systems,  these  same 
factors  would  be  concerned  also  in  certain  other  properties  of  optical 
images  (brightness,  resolving-power,  etc.)  besides  those  above-men- 
tioned. Throughout  this  chapter  it  will  be  tacitly  assumed  (unless 
expressly  stated  otherwise)  that  the  optical  system  is  free  from  aber- 
rations both  spherical  and  chromatic,  so  that  we  have  to  do  with  the 
case  of  so-called  GAUssian  Imagery,  without  colour-faults.  Of  course, 
it  must  be  borne  in  mind  that  the  formulae  and  results  derived  on 
these  assumptions  will  have  to  be  applied  with  due  caution  to  the  more 
or  less  imperfect  imagery  that  is  realized  in  the  case  of  actual  optical 

532 


§  361.]  The  Aperture  and  the  Field  of  View.  533 

instruments;  but,  on  the  other  hand,  it  would  lead  us  too  far  and  tend 
only  to  confuse  the  matter  in  hand  if  we  attempted  here  to  go  into 
all  the  intricate  and  special  questions  that  are  involved  when  the  dif- 
ferent aberrations  are  taken  into  account. 

The  bundles  of  rays  that  traverse  an  optical  instrument  are  limited 
either  by  the  physical  dimensions  of  the  lenses  themselves  or  by  per- 
forated diaphragms  or  "stops"  interposed  specially  for  this  purpose. 
In  all  cases  that  possess  interest  for  us  such  stops  are  circular  in  form 
and  concentric  with  the  optical  axis.  The  direct  and  obvious  effect 
of  a  stop  or  lens-rim  is  two-fold,  viz.,  first,  to  restrict  the  apertures  of 
the  bundles  of  effective  rays,  and,  second,  to  limit  the  extent  of  the  object 
that  is  reproduced  in  the  image.  The  mode  and  measure  of  these  re- 
strictions will  depend  on  the  sizes  and  positions  of  the  stops  and  also 
on  the  type  of  the  optical  apparatus  itself. 

361.    The  Aperture-Stop. 

In  the  general  and  at  the  same  time  the  most  usual  case,  the  dia- 
phragm or  stop  is  placed  with  its  centre  on  the  optical  axis  at  some  point 
lying  between  two  consecutive  lenses  of  the  optical  system  L;  which 
is  thereby  divided  into  two  parts,  a  front  component  (Lt)  consisting 
of  the  part  of  the  lens-system  in  front  of  the  interior  stop,  and  a  hinder 
component  (L2)  consisting  of  the  remainder  of  the  lens-system  lying 
on  the  other  or  far  side  of  the  stop.  There  may  be  also  not  merely 
one  but  several  such  interior  stops,  either  actual  perforated  diaphragms 
or  the  rims  of  the  lenses  themselves;  each  of  which,  according  to  its 
position,  will  divide  the  lens-system  into  two  parts,  as  above-men- 
tioned. Frequently  a  stop  is  placed  in  front  of  the  entire  system,  in 
which  case  it  is  called  a  front  stop.  And,  similarly,  a  stop  which  is 
placed  behind,  or  towards  the  image-side  of,  the  optical  system  (as  is 
also  not  uncommon)  is  called  a  rear  stop.  With  respect  to  a  front 
stop,  L2  =  L,  and  with  respect  to  a  rear  stop,  Ll  =  L. 

The  apertures  of  the  bundles  of  effective  rays  are  conditioned  by 
these  stops.  In  the  simplest  case  of  all  when  the  optical  system  con- 
sists of  a  single  lens  whose  two  surfaces  intersect  in  the  circular  rim 
of  the  lens,  this  circle  is  the  common  base  of  the  cones  of  incident 
and  refracted  rays  that  take  part  in  the  image-phenomena;  and  here 
the  bundles  of  effective  rays  are  limited  by  the  surface  of  the  lens  itself. 

If  now  we  interpose  between  the  axial  object-point  M  (Fig.  157) 
and  the  lens  a  front  stop  with  its  centre  on  the  axis  at  the  point  desig- 
nated in  the  figure  by  M  whose  diameter  subtends  at  M  an  angle  smaller 
than  that  subtended  at  the  same  point  by  the  diameter  of  the  lens, 
this  stop  will  evidently  limit  the  aperture  of  the  bundle  of  object- 


534 


Geometrical  Optics,  Chapter  XIV. 


[  §  361. 


rays  emanating  from  the  axial  object-point  M ;  and  if  the  position 
on  the  axis  of  the  point  which  is  conjugate  to  M  is  designated  by  M't 
the  GAUSsian  image  of  the  circular  stop  in  the  transversal  plane  <r 
of  the  Object-Space  with  its  centre  at  M  will  be  a  circle  with  its  centre 
at  M'  lying  in  the  transversal  plane  <r'  conjugate  to  <r.  Since  all  the 


INFINITELY  THIN  CONVEX  I,ENS  WITH  FRONT  STOP.  CD  is  the  Aperture-Stop  with  its  centre 
on  the  optical  axis  at  M,  CD  is  here  also  the  Entrance-Pupil  ;  C  '  Lf  the  Exit-Pupil.  M'tf  is  the 
image  of  the  object  MQ. 


rays  that  before  refraction  go  through  the  stop  at  M  must  after  re- 
fraction pass  through  the  stop-image  at  AT,  we  see  that,  whereas  the 
material  stop  placed  in  front  of  the  lens  at  M  limits  the  apertures  of 
the  bundles  of  effective  rays  in  the  Object-Space,  the  stop-image  at 
M'  performs  the  same  office  for  the  bundles  of  rays  in  the  Image-Space; 
or,  in  other  words,  the  front  stop  lying  in  the  transversal  plane  <r  is 
the  common  base  of  all  the  cones  of  object-rays,  and,  similarly,  the 
stop-image  in  the  transversal  plane  <r'  is  the  common  base  of  all  the 
cones  of  image-rays. 

1  Proceeding  now  to  the  most  general  case,  let  us  suppose  that  the 
optical  system  L  is  composed  of  several  lenses  and  provided  with 
one  or  more  interior  stops,  either  perforated  diaphragms  or  lens- 
rims.  We  begin  by  constructing  the  GAUSsian  image  of  each  stop  0 
(Fig.  158)  formed  by  that  part  Lt  of  the  system  that  lies  in  front  of 
(or  to  the  left  of)  0.  The  stop  that  corresponds  to  that  one  of  these 
images  that  subtends  the  smallest  angle  at  the  selected  axial  object- 
point  M  is  called  the  aperture-stop;  because  this  is  evidently  the  stop 
that,  with  respect  to  Af,  conditions  the  apertures  of  the  bundles  of 


§  361.] 


The  Aperture  and  the  Field  of  View. 


535 


effective  rays.  In  the  figure  the  aperture-stop  is  represented  as  the 
one  with  its  centre  located  at  the  point  0,  whose  image  formed  by 
the  front  part  Ll  of  the  optical  system  L  in  the  transversal  plane  cr 
that  is  crossed  by  the  axis  at  the  point  M  subtends  a  smaller  angle  at 
M  than  the  corresponding  image  of  any  of  the  other  stops.  Which 
one  of  the  perforated  diaphragms  or  lens-rims  plays  the  r61e  of  aperture- 
stop  will  depend  essentially  on  the  position  of  the  axial  object-point  M . 


FIG.  158. 


COMPOUND  OPTICAL  SYSTEM  CONSISTING  OF  Two  THIN  I^ENSES  L\,  Za,  SEPARATED  BY  INTERIOR 
APERTURE-STOP  WITH  CENTRE  AT  O.  The  axial  point  Jf  conjugate  to  O  with  respect  to  L\  and  the 
axial  point  M'  conjugate  to  O  with  respect  to  Lz  (and  therefore  conjugate  also  to  M  with  respect 
to  Li  +  Lz)  are  the  centres  of  the  Entrance-Pupil  and  Exit-Pupil,  respectively.  M'Q'  is  the  image 
of  the  object  MQ. 


MM 


=  p. 


L  MMD  - 


In  passing,  it  may  be  observed  that  a  case  may  occur,  such  as  that 
shown  in  Fig.  159,  in  which  the  images  of  two  (or  more)  of  the  material 
stops  formed  by  the  parts  of  the  optical  system  lying  in  front  of  them 
subtend  at  the  axial  object-point  M  angles  of  equal  magnitude; 
so  that  (if  this  angle  is  also  the  smallest  of  all  such  angles)  either  of 
these  two  stops  may  be  regarded  as  the  aperture-stop.  The  point 
of  intersection  of  the  pair  of  straight  lines  joining  the  upper  extremity 
of  one  stop-image  with  the  lower  extremity  of  the  other  determines  a 
second  point  K  on  the  optical  axis  at  which  the  two  stop-images  also 
subtend  angles  of  equal  magnitude.  With  respect  to  an  axial  object- 
point  situated  anywhere  between  the  two  extreme  positions  M  and 
K,  the  stop-image  marked  77  in  the  diagram  will  subtend  the  smaller 
angle  of  the  two;  whereas  for  an  axial  object-point  lying  anywhere 
outside  the  segment  M  K  the  stop-image  marked  /  will  subtend  the 


536 


Geometrical  Optics,  Chapter  XIV. 


[§361. 


smaller  angle.1  It  is  apparent  that  the  stop  that  acts  as  the  aperture- 
stop  for  an  object  in  one  position  on  the  axis  may  not  be  the  aperture- 
stop  for  another  position  of  the  object.  We  must  assume,  therefore, 
that  the  object  has  a  fixed  position  or  at  any  rate  that  it  is  movable 
within  certain  prescribed  limits  if  the  stops  are  to  retain  their  functions, 
as  is  necessary,  for  example,  in  the  case  of  such  optical  instruments  as 
the  telescope  and  the  microscope. 

Returning  to  the  consideration  of  Fig.  158,  we  see  that,  since  the 
aperture-stop  at  0  must  be  the  common  base  of  all  the  cones  of  rays 
after  their  emergence  from  the  front  part  L^  of  the  optical  system,  the 
stop-image  in  the  transversal  plane  <r  must  likewise  be  the  common 
base  of  all  the  cones  of  rays  in  the  Object-Space.  Moreover,  if  M' 
designates  the  position  of  the  point  which,  with  respect  to  the  hinder 
part  L2  of  the  optical  system,  is  conjugate  to  the  stop-centre  0,  the 


FIG. 159. 
CASE  OF  Two  ENTRANCE-PUPILS. 

image  of  the  stop  formed  by  L2  will  lie  in  the  transversal  plane  <r' 
determined  by  the  axial  point  M';  and,  similarly,  this  stop-image  will 
evidently  be  the  common  base  of  all  the  cones  of  image-rays  after 
having  traversed  the  entire  compound  system  L  =  L^  +  L2.  Evi- 
dently, also,  the  transversal  planes  or,  or'  are  a  pair  of  conjugate  planes, 
so  that  the  stop-images  at  M  and  M '  are  images  of  each  other  with 
respect  to  the  whole  system  L.  Together  they  constitute  a  pair  of 
virtual  stops  (as  distinguished  from  actual  or  material  stops)  that  are 
the  measures  of  the  apertures  of  the  ray-bundles  in  the  Object-Space 
and  Image-Space.  A  material  stop  of  the  same  size  and  position  as 

1  See  M.  VON  ROHR:  "  Die  Strahlenbegrenzung  in  optischen  Systemen  ",  Chapter  IX 
of  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR. 
See  p.  469. 


§  362.]  The  Aperture  and  the  Field  of  View.  537 

the  stop-image  at  M  will  act  exactly  in  the  same  way  with  respect  to 
the  limiting  of  the  bundles  of  rays  as  a  material  stop  identical  in  size 
and  position  with  the  stop-image  at  M' ';  and  either  of  them  or  both 
together,  so  far  as  this  effect  is  concerned,  would  be  precisely  equiva- 
lent to  the  actual  stop  that  we  suppose  to  be  situated  at  0.  ABBE/ 
who  has  done  most  to  develop  the  theory  of  stops,  calls  the  stop-images 
at  M  and  M',  from  an  analogy  with  the  optical  system  of  the  human 
eye,  the  pupils  of  the  system.  The  pupil  of  the  eye  is  the  contractile 
aperture  of  the  iris,  the  image  of  which  produced  by  the  cornea  and 
the  aqueous  humour  lies  in  front  of  the  eye  (as  can  be  seen  by  looking 
directly  into  the  eye) ;  so  that  only  such  rays  as  are  directed  towards 
this  image  can  enter  the  eye  through  the  iris-opening.  From  this 
same  analogy,  ABBE  calls  also  the  aperture-stop  at  0  the  iris  of  the 
optical  system.  The  two  pupils  at  M  and  M'  are  distinguished  by 
the  names  Entrance- Pupil  and  Exit- Pupil,  respectively.2 

362.  An  imagery  is  completely  determined  so  soon  as  we  know  the 
positions  of  the  two  pairs  of  conjugate  transversal  planes  cr,  a'  and  <r,  <r', 
together  with  the  values  of  the  magnification-ratios  Y  and  Y  that 
characterize  these  two  pairs  of  planes.  Thus,  if  the  pupils  of  the  sys- 
tem are  given  in  both  size  and  position,  and  if  also  the  image  M'Q' 
corresponding  to  a  given  object-line  M Q  at  right  angles  to  the  optical 
axis  has  been  constructed,  the  procedure  of  every  ray  that  traverses 
the  system  can  be  ascertained  immediately  without  taking  farther  ac- 
count of  the  special  construction  of  the  apparatus.  For  example,  to 
an  object-ray  QD  which,  originating  at  the  object-point  Q  crosses  the 
(r-plane  at  a  point  D  in  the  circumference  of  the  entrance-pupil  there 
must  correspond  an  image-ray  directed  toward  the  image-point  Q'  and 
going  through  the  point  D'  of  the  circumference  of  the  exit-pupil  that 
is  conjugate  to  the  point  D.  It  is  evident  also  that  the  totality  of  the 
effective  rays  in  the  Object-Space  may  be  regarded  in  either  of  two 
ways,  viz.:  (i)  As  cones  of  rays  emanating  from  points  in  the  object 
MQ  and  having  the  entrance-pupil  as  a  common  cross-section;  or 
(2)  As  cones  of  rays  with  their  vertices  at  points  of  the  entrance-pupil 
and  a  common  base  in  the  object  MQ;  so  that  the  roles  of  object  and 
entrance-pupil  are  interchangeable.  This  same  reciprocity  exists  like- 
wise between  the  image  and  the  exit-pupil. 

1  See  E.  ABBE:  Beitraege  zur  Theorie  des  Mikroskops  und  der  mikroskopischen  Wahr- 
nehmung:  Archiv  f.  mikr.  Anat.,  ix  (1873),  413-468.     Also,  Gesammelte  Abhandlungen, 
Bd.  I  (Jena,  1904),  4S~ioo. 

2  See  also  E.  ABBE:  Ueber  die  Bestimmung  der  Lichtstaerke  optischer  Instrumente: 
Jen.  Zft.f.  Med.  u.  Naturiv.,  vi  (1871),  263-291.     Also,  Gesammelte  Abhandlungen,  Bd.  I 
(Jena,  1904).  14-44- 


538  Geometrical  Optics,  Chapter  XIV.  [  §  364. 

363.     The  Aperture-  Angle. 

The  angle  MMD  =  0,  denned  more  precisely  by  the  relation 

MD 


where  D  designates  the  position  of  a  point  in  the  meridian  plane  lying 
in  the  circumference  of  the  entrance-pupil,  is  called  the  aperture-angle 
of  the  optical  system.  If  p  =  MD  denotes  the  radius  of  the  entrance- 
pupil  (reckoned  positive  or  negative  according  as  D  lies  above  or  below 
the  optical  axis),  and  if  MM  =  £  denotes  the  abscissa  of  the  axial 
object-point  M  with  respect  to  the  centre  M  of  the  entrance-pupil  as 
origin,  we  may  write  : 

tan0=-|.  (442) 

Similarly,  if  9'  =  Z  Af'Af'ZX,  p'  =  M'D',  £'  =  M'M',  where  the  points 
designated  by  M',  M',  D'  are  conjugate  to  the  points  in  the  Object- 
Space  designated  by  the  same  letters  without  the  primes,  we  have  also  : 

jj 
tan  0'=-  |r.  (443) 

364.    The  Numerical  Aperture. 

Although  the  size  of  the  aperture-angle  0  is  in  a  certain  more  or  less 
geometrical  sense  a  measure  of  the  number  of  effective  rays  emanating 
from  the  axial  object-point  M,  this  angle  by  itself,  from  an  optical 
standpoint,  is  not  a  true  criterion  of  the  aperture  of  the  optical  sys- 
tem. All  the  rays  of  a  bundle  are  not  of  equal  optical  value,  and  on 
this  account  the  quantity  of  light-energy  that  is  transmitted  through 
the  optical  system  from  an  object-point  to  its  conjugate  image-point 
depends  on  something  more  than  just  the  size  of  the  aperture  angle. 
A  luminous  surface-element  emits  more  energy  along  some  directions 
than  along  others,  the  intensity  of  radiation  (§  388),  according  to  LAM- 
BERT'S law,  being  proportional  to  the  cosine  of  the  angle  of  emission  ;  so 
that  the  most  energetic  ray  is  the  one  that  is  directed  along  the  normal 
to  the  surface-element  at  the  origin-point  of  the  ray.  Consequently 
different  rays  emanating  from  the  same  object-point  will  be  the  routes 
through  the  entrance-pupil  of  the  optical  system  of  different  cargoes 
of  light-energy. 

According  to  ABBE,1  the  proper  and  rational  measure  of  the  aperture 

1  E.  ABBE:  Die  optischen  Huelfsmittel  der  Mikroskopie:  Gesammelte  Abhandlungen, 
Bd.  I  (Jena,  1904),  1  19-164  j  especially,  p.  142.  (This  paper  was  published  originally 


§  364.]  The  Aperture  and  the  Field  of  View.  539 

of  an  optical  system  —  the  only  one  indeed  that  affords  a  just  idea  of 
its  efficiency  —  is  given  by  the  product  of  the  refractive  index  of  the 
first  medium  (n)  and  the  sine  of  the  aperture-angle;  this  pif)duct,  to 
which  ABBE  gives  the  name  numerical  aperture,  and  which  is  denoted 
here  by  the  symbol  A  ,  has  therefore  the  following  expression  : 

A  =  n-sin.  0.  (444) 

It  would  derange  too  much  the  plan  of  this  treatise  if  we  paused  here 
to  explain  fully  the  basis  of  this  definition,  especially  also  as  such  an 
exposition  belongs  rather  to  the  special  theory  of  optical  instruments 
and  to  the  theory  of  the  microscope  in  particular  where  the  numerical 
aperture  has  an  exceedingly  important  role.  In  the  case  of  the  instru- 
ment just  mentioned,  the  conjugate  axial  points  M,  M'  are  the  apla- 
natic  pair  of  points  of  the  optical  system  (§  279),  and  under  these  cir- 
cumstances it  would  be  easy  to  show  that  the  quantity  of  radiant  energy 
transmitted  from  M  to  Mf  is  proportional  to  the  numerical  aperture. 

It  may  be  remarked  that  the  magnitude  denoted  by  A  is  propor- 
tional, not  to  the  aperture-angle  0,  but  to  the  sine  of  this  angle;  so 
that,  for  example,  if  0  were  increased  from,  say,  30°  to  90°,  the  numeri- 
cal aperture  would  be  only  doubled,  since  sin  90°  :  sin  30°  =  2  :  I.  The 
numerical  aperture  is  also  proportional  to  the  refractive  index,  so  that 
its  value  can  be  altered  merely  by  immersing  the  object  in  a  different 
medium  for  which  n  has  a  different  value;  and,  hence,  as  ABBE  has 
observed,  this  measure  A  enables  us  to  compare  the  apertures  of  the 
so-called  "dry"  and  "immersion"  optical  systems. 

The  relation  between  the  numerical  aperture  and  the  radius  (p) 
of  the  entrance-pupil  and  the  abscissa  £  =  MM  is  exhibited  by  the 
formula  : 


whence  also  we  can  see  the  effect  on  the  aperture  of  a  displacement 
5£  of  the  object-point  M.  Whether  the  aperture  will  be  increased  or 
diminished  by  such  a  variation  of  the  position  of  the  axial  object- 
point  Mj  will  depend  on  the  signs  of  both  £  and  5£. 

If  Z  denotes  the  angular  magnification  of  the  system  with  respect 

In  Braunschweig  in  1878.)     Also: 

E.  ABBE:  Ueber  die  Bedingungen  des  Aplanatismus  der  Linsensysteme:  Silzungsber. 
d.  Jen.  Gesellschaft  f.  Med.  u.  Naturw.,  1879,  129-142.  See  Gesammelte  Abhandlungen, 
Bd.  I  (Jena,  1904),  213-226;  especially,  pages  225  and  226.  Also: 

E.  ABBE:  On  the  Estimation  of  Aperture  in  the  Microscope:  Journ.  Roy.  Micr.  Soc., 
(2),  i  (1881),  388-423:  especially,  pages  395  &  396.  (A  German  translation  of  this 
paper  is  in  Gesammelte  Abhandlungen,  Bd.  I,  325-374.) 


540  Geometrical  Optics,  Chapter  XIV.  [  §  365. 

to  the  pupil-centres  M ,  M ',  and  if  F  denotes  the  lateral  magnification 
with  respect  to  the  pair  of  conjugate  points  M,  M',  then,  according 
to  the  last  of  the  image-equations  (127),  we  shall  have: 

t'      V 

f-|;  (446) 

and,  hence,  in  the  special  case  when  the  points  M,  Mf  are  the  aplanatic 
pair  of  points,  so  that 

n  •  sin  6        A 

n'-sine'  =  Z'         ' 
we  obtain  the  relation : 

•       "      j'"fZ;  (447) 

which  will  be  found  to  be  a  very  useful  formura  in  the  special  theory 
of  optical  instruments. 

ART.    115.     THE   CHIEF   RAYS   AND   THE   RAY-PROCEDURE. 

365.  Chief  Ray  as  Representative  of  Bundle  of  Rays.  The  rays 
which,  emanating  from  all  the  points  of  the  object,  are  directed 
towards  the  centre  M  of  the  entrance-pupil  constitute  the  bundle  of 
so-called  chief  rays  in  the  Object-Space;  to  which  in  the  Image- 
Space  there  corresponds  also  a  conjugate  bundle  of  chief  rays  which 
all  meet  at  the  centre  Mr  of  the  exit-pupil.  Accordingly,  the  pupil- 
centres  M,  M '  are  to  be  considered  as  the  centres  of  perspective  of  the 
Object-Space  and  Image-Space,  since  to  any  object-point  P  lying  on 
the  chief  object-ray  PM  there  corresponds  an  image-point  Pr  lying 
on  the  conjugate  chief  image-ray  P'M' .  The  chief  ray  is  the  axis 
of  symmetry  of  the  cone  of  rays,  and,  therefore,  especially  when  the 
circular  aperture-stop  is  very  small,  it  may  be  regarded  as  the  repre- 
sentative ray  of  the  bundle  (cf.  §  286) ;  and,  hence,  a  knowledge  of  the 
procedures  of  the  chief  rays  will  often  afford  an  accurate  idea  of  the 
entire  image-process. 

Since  the  pupil-centres  are  the  centres  of  perspective  of  the  Object- 
Space  and  Image-Space,  object-points  which  lie  along  a  chief  ray  in 
the  Object-Space  will  be  reproduced  by  image-points  which  lie  along 
the  conjugate  chief  ray  in  the  Image-Space,  and  which,  therefore,  if 
viewed  by  an  eye  placed  at  the  exit-pupil  (which  is  the  usual  place  for 
the  eye  in  order  that  the  entire  image  may  be  all  commanded  at  the 
same  time),  will  appear  to  lie  all  at  the  same  place.  If  the  image  is 
received  on  a  plane  screen,  placed  at  right  angles  to  the  optical  axis, 


366.] 


The  Aperture  and  the  Field  of  View. 


541 


and  if  this  screen  does  not  coincide  exactly  with  the  transversal  image- 
plane  or'  which  is  conjugate  to  the  transversal  plane  o-  in  the  Object- 
Space  that  contains  the  object-point  Q  (Fig.  160),  the  image  of  Q  on 


FIG.  160. 


BLUR-CIRCLES  IN  THE  SCREEN-PLANE  DUE  TO  IMPERFECT  FOCUSSING,  xx?  is  the  optical  axis 
of  the  system  L.  CD,  C'/X  diameters  of  Entrance-Pupil  and  Exit-Pupil.  M'Cf  is  the  image  of 
MQ,  and  M"  and  Q"  are  the  centres  of  the  blur-circles  in  the  Screen-Plane  corresponding  to  the 
object-points  M  and  Q,  respectively. 


MQ=y,    M'Q'=yf,    M"Q"=y", 


=  p.  M  '//=/>' 

L  M' M'Lf  =  ©'. 


the  screen  will  not  be  a  point  but  an  aberration-figure  coinciding  with 
the  section  of  the  bundle  of  image-rays  made  by  the  screen-plane.  If 
the  aperture-stop  is  circular  in  form,  this  aberration-figure  will  be  a 
circle  (so-called  "blur-circle"),  and  the  centre  of  the  circle  where  the 
chief  ray  crosses  the  screen-plane  will  be  regarded  as  the  place  on  the 
screen  of  the  image  corresponding  to  the  object-point  Q.  The  smaller 
the  diameter  of  the  exit-pupil,  the  smaller  will  be  the  diameter  of  the 
blur-circle;  and  if  the  diameter  of  the  aperture-stop  is  infinitely  small, 
the  blur-circles  will  all  contract  into  points  at  their  centres. 

366.     Optical  Measuring  Instruments. 

The  importance  of  taking  into  consideration  the  procedures  of  the 
chief  rays  may  be  illustrated  by  investigating  the  class  of  optical  in- 
struments that  are  especially  contrived  for  determining  the  size  of  an 
pbject  by  measuring  the  size  of  the  image.  The  image  may  be  cast 
on  a  screen  which  is  provided  with  a  scale  or  the  image  may  be  formed 
in  the  air  in  a  plane  containing  a  material  scale  or  a  scale-image.  But 
here,  owing  partly  perhaps  to  the  unavoidable  dioptric  imperfections 
of  the  image  itself  but  above  all  to  the  difficulty  of  focussing  the  instru- 
ment exactly  so  that  the  true  image-plane  coincides  with  the  scale- 
plane,  there  is  a  source  of  error  in  the  method,  since,  instead  of  measur- 


542 


Geometrical  Optics,  Chapter  XIV. 


[  §  366. 


ing  the  size  of  the  true  image,  we  may  be  measuring  that  of  the  apparent 
and  more  or  less  blurred  image  as  viewed  by  the  eye  in  the  scale- 
plane.  The  size  of  this  image  in  the  scale-plane  is  determined  by  the 
procedures  of  the  chief  rays. 

The  steeper  the  slopes  of  the  chief  image-rays,  the  greater  will  be 
the  error  due  to  imperfect  focussing.  If,  for  example,  the  centre  0  (Fig. 
161)  of  the  aperture-stop  (§  361)  coincides  with  the  primary  focal  point 
F2  of  the  posterior  part  L2  of  the  optical  system  L,  the  centre  M'  of 
the  exit-pupil  will  be  the  infinitely  distant  point  of  the  optical  axis, 
and  hence  the  parallactic  error  in  the  measurement  of  the  size  of  the 
image  will  vanish  entirely.  In  an  optical  measuring  instrument,  such 
as  the  micrometer-microscope,  in  which  the  plane  of  the  cross-hairs 
or  the  scale-plane  has  a  fixed  position,  while  the  distance  of  the  object 


FIG.  161. 

OPTICAL  SYSTEM  TELECENTRIC  ON  THE  SIDE  OF  THE  IMAGE.  The  centre  O  of  the  Aperture- 
Stop  is  at  the  primary  focal  point  Fz  of  the  lyens-System  Lz  l  so  that  the  centre  M  '  of  the  Exit-Pupil 
is  the  infinitely  distant  point  of  the  optical  axis  xx*  '.  The  blurred  image  M"  Q"  of  the  object  MQ 
is  of  the  same  size  as  the  true  image  M'Q'  ,  no  matter  how  we  adjust  the  screen-plane  a"  , 


yf  . 


=  y,    M'  Q'  =  M' 
=  p.    M'D'=p't 


is  adjustable,  we  must  contrive  therefore  so  that  the  exit-pupil  will  be 
infinitely  distant,  in  which  case  the  image  will  appear  of  the  same  size 
whether  it  lies  in  the  scale-plane  or  not.  On  the  other  hand,  if  the 
object  is  fixed  and  the  position  of  the  scale-plane  variable  (as,  for 
example,  in  the  case  of  a  telescope  in  which  the  object  is  usually  im- 
movable and  the  adjustment  is  accomplished  by  the  eye-piece),  the 
endeavour  is  to  contrive  so  that  the  distance  of  the  object  from  the 
entrance-pupil  will  have  no  effect  on  the  measurement  of  the  lateral 
magnification  Y  =  yf/y\  which  can  be  achieved  in  a  similar  way  by 
designing  the  instrument  so  that  the  centre  0  of  the  aperture-stop 
coincides  with  the  secondary  focal  point  E{  of  the  part  Lv  of  the 
optical  system  that  lies  in  front  of  it,  whereby  the  centre  M  of  the 


§  367.]  The  Aperture  and  the  Field  of  View.  543 

entrance-pupil  will  be  the  infinitely  distant  point  of  the  optical  axis, 
and  the  chief  rays  in  the  Object-Space  will,  therefore,  be  parallel  to 
the  axis. 

An  optical  system  in  which  the  centre  0  of  the  aperture-stop  coin- 
cides with  one  or  other  of  the  two  focal  points  that  are  here  designated 
by  E\  and  F2  is  called  by  ABBEI  a  telecentric  system.  According  as 
it  is  the  entrance-pupil  or  the  exit-pupil  which  is  the  infinitely  distant 
one  of  the  two  pupils,  the  system  is  said  to  be  "telecentric  on  the  side 
of  the  object"  or  "telecentric  on  the  side  of  the  image",  respectively. 
In  the  special  case  when  the  focal  points  E[  and  F2  coincide  with  each 
other  the  system  will  be  telescopic  (§  186,  Case  i);  and  if,  moreover, 
the  centre  0  of  the  aperture-stop  coincides  with  both  of  these  focal 
points,  the  system  will  be  "telecentric  on  both  sides". 

367.  If  the  positions  of  the  two  focal  points  of  the  optical  system 
are  designated  by  F  and  E',  and  if  the  magnitudes  of  the  focal  lengths 
are  denoted  by  /  and  e',  and  if,  finally,  x  =  FM,  x'  =  E'M'  denote 
the  abscissae,  with  respect  to  the  focal  points,  of  the  pair  of  conjugate 
axial  points  M,  M'-,  then,  on  the  assumption  of  perfect  collinear  cor- 
respondence, we  have,  according  to  the  second  of  formulae  (i  15),  for  the 
lateral  magnification  of  the  system  with  respect  to  the  points  M,  M': 


-- 
-X~   e" 

In  the  special  case,  therefore,  when  the  centre  M  of  the  entrance-pupil 
coincides  with  the  position  F  of  the  primary  focal  point,  so  that 
x  =  FM  =  MM  =  £,  we  obtain: 


and,  hence,  when  the  system  is  telecentric  on  the  side  of  the  image, 
the  magnification  Y  will  not  depend  on  the  position  of  the  scale-plane, 
but  only  on  the  position  of  the  object-plane  a.  Similarly,  when  the 
centre  M  '  of  the  exit-pupil  coincides  with  the  position  E'  of  the  sec- 
ondary focal  point  (*'  =  E'M'  =  M'M'  =  £'),  we  find: 


which  shows  that  when  the  system  is  telecentric  on  the  side  of  the 

*E.  ABBE:  Ueber  mikrometrische  Messung  mittelst  optischer  Bilder:  Sitzungsber. 
d.  Jen.  Gesellschaft  f.  Med.  u.  Nalurw.,  1878,  11-17.  See  also:  Gesammelte  Abhandhingen, 
Bd.  I  (Jena.  1904),  165-172. 


544  Geometrical  Optics,  Chapter  XIV.  [  §  368. 

object,  the  magnification  Y  is  independent  of  the  position  of  the  object- 
plane  cr. 

ART.    116.     MAGNIFYING   POWER. 

368.     The  Objective  Magnifying  Power. 

If  the  optical  image  is  received  upon  a  screen,  as  in  the  case  of  the 
image  produced  by  the  photographic  objective  or  the  projection-micro- 
scope, or  if  the  dimensions  of  the  image  suspended  in  air  are  to  be  de- 
termined by  comparison  with  a  scale  (optical  measuring  instrument, 
§  366),  the  lateral  magnification 


defined  by  the  ratio  of  the  corresponding  linear  dimensions  y,  y'  of 
object  and  image,  which,  theoretically  at  least,  as  we  have  seen  (§  179), 
may  have  any  value  depending  on  the  position  of  the  axial  object- 
point  M,  is  the  measure  of  the  actual  or  so-called  objective  magnifica- 
tion. 

In  case  the  object  MQ  =  y  is  very  distant  or  inaccessible,  instead 
of  comparing  the  actual  dimensions  of  the  image  on  the  screen  with 
those  of  the  object,  we  may  have  to  measure  the  magnifying  power  of 
the  optical  projection-system  in  some  other  way,  for  example,  by  means 
of  the  ratio  of  the  linear  size  of  the  image  (M'Qf  =  y')  to  the  apparent 
size  of  the  object;  which  latter  expression  needs  however  to  be  ex- 
plained more  precisely.  By  the  apparent  size  of  the  object  is  meant 
here  the  angle  (or  the  trigonometric  tangent  of  the  angle)  MMQ  =  9 
subtended  by  the  object  at  the  centre  M  of  the  entrance-pupil,  which 
is  also  the  same  thing  as  the  slope-angle  of  the  chief  ray  proceeding 
from  the  extra-axial  object-point  Q.  This  ratio 


tan0 

is  always  a  more  or  less  important  factor  in  the  determination  of  the 
objective  magnifying  power  of  an  optical  projection-system.  In  the 
limiting  case  when  the  object  is  infinitely  distant,  and  the  image  is 
formed  therefore  in  the  secondary  focal  plane  of  the  optical  system, 
the  size  of  the  image  y'  =  M'Q'  =  E'Q'  (since  the  axial  image-point 
Mf  coincides  with  the  secondary  focal  point  £')  will  depend  only  on 
the  slope  8  of  the  cylindrical  bundle  of  object-rays  coming  from  the 
infinitely  distant  object-point  <2;  so  that  under  these  circumstances  the 
ratio  above  defined  is  the  real  measure  of  the  objective  magnification 
produced  by  the  optical  projection-system. 


§  369.]  The  Aperture  and  the  Field  of  View.  545 

Since 

tan0  =  |, 

where  £  =  MM  denotes  the  abscissa  of  the  axial  object-point  M  with 
respect  to  the  centre  M  of  the  entrance-pupil,  and  since,  moreover, 

/ 


where  x  =  FM  and  x  =  FM  denote  the  abscissae,  with  respect  to  the 
primary  focal  point  F,  of  the  points  M  and  M  ,  respectively,  and  where 
/  denotes  the  primary  focal  length  of  the  optical  system;  we  obtain 
finally  : 

=A~/-  (448) 

If  the  object  is  at  a  great  distance,  x  will  be  very  small  compared  with 
£,  so  that  the  fraction  £/(*  -f-  f)  will  be  very  nearly  equal  to  unity; 
and  hence  we  may  write  : 

y 

£^0  =  /,  approximately; 

which  will  be  not  only  approximately  but  strictly  true  in  case  either 
the  object  is  infinitely  distant  (£  =  oo)  or  the  plane  <r  of  the  entrance- 
pupil  coincides  with  the  primary  focal  plane  (:c  =  o). 

In  making  geodetic  measurements  it  often  happens  that  one  wishes 
to  determine  the  distance  of  the  object  (a  surveyor's  rod,  for  example) 
by  measuring  the  size  of  its  image.  If  the  entrance-pupil  of  the  optical 
instrument  is  situated  in  the  primary  focal  plane,  the  angle  0  can  be 
determined  by  the  relation  found  above: 

V 
tan  6  =  j  , 

and  hence  the  distance  of  the  object  may  be  computed  by  the  formula  : 

'i'm    y     -f  2- 

*~tanQ~J'  y" 

provided  we  know  the  values  of  the  magnitudes  denoted  by  y,  y'  and  /. 

369.    The  Subjective  Magnifying  Power. 

If,  however,  the  optical  instrument  is  designed  to  be  used  subject- 
ively in  conjunction  with  the  eye  for  the  purpose  of  reinforcing  vision 
36 


546  Geometrical  Optics,  Chapter  XIV.  [  §  369. 

(as,  for  example,  in  the  case  of  the  ordinary  magnifying  glass  or  the 
microscope,  etc.),  the  magnitude  of  the  image  produced  by  the  in- 
strument will  not,  by  itself,  serve  as  a  measure  of  the  magnification, 
but  the  question  here  is  rather  with  respect  to  the  size  of  the  image  that 
is  formed  on  the  retina  of  the  eye;  and  since  this  latter  cannot  be  sub- 
jected to  direct  measurement,  the  formula  F  =  y'/y  (where  y'  denotes 
the  size  of  the  retina-image)  is  not  applicable  in  this  case. 

In  the  ordinary  acceptation  of  the  term,  the  subjective  magnifying 
power  of  an  optical  instrument  which,  as  in  the  case  of  the  microscope, 
is  to  be  used  in  conjunction  with  the  eye,  is  not  the  ratio  of  the  actual 
dimensions  of  the  image  and  object,  but  the  ratio  of  their  apparent 
sizes  as  seen  by  the  eye  of  the  observer.  Here  also  it  is  necessary  to 
explain  distinctly  what  is  meant  by  the  "apparent  size"  of  both  the 
image  and  the  object. 

The  apparent  size  of  the  image  M'Q'  =  y'  as  viewed  by  an  eye 
placed  at  a  point  J  on  the  axis  of  the  optical  system  is  measured  by 
the  angle  (or,  rather,  by  the  tangent  of  the  angle)  subtended  at  J  by 
M'Q'  ',  that  is,  by 


The  place  of  the  eye  here  designated  by  /  is  actually  the  centre  of  the 
entrance-pupil  of  the  eye,  which  is  usually  placed  so  as  to  coincide 
with  the  centre  M  '  of  the  exit-pupil  of  the  optical  system.  If  this  is 
the  adjustment,  then  JM'  =  M'M'  =  £',  and 

LM'JQ  =  LM'M'Q'  =  6', 

where  8'  denotes  the  slope  of  the  chief  image-ray  of  the  bundle  of 
rays  that  go  through  the  image-point  Q'.  Introducing  these  symbols, 
we  obtain  the  following  expression  for  the  apparent  size  of  the  image  : 

y 

tear-  |r> 

If  now  the  optical  instrument  is  removed,  and  the  eye  is  focussed 
so  as  to  view  directly  and  distinctly  the  object  M  Q  =  y,  the  apparent 
size  of  the  object  as  viewed  by  the  eye  at  the  distance  of  distinct 
vision  JM  =  a  is  measured  by  the  angle  MJQ  =  rjy  that  is,  by 

y 

tan  i]  =  -  . 
a 

Thus,  according  to  the  usual  definition,  what  is  meant  by  the  subjective 


§  369.]  The  Aperture  and  the  Field  of  View.  547 

magnifying  power  of  an  optical  instrument  belonging  to  the  same 
general  class  as  the  microscope  is  the  ratio  of  the  visual  angles  (or 
trigonometric  tangents  of  the  angles)  subtended  at  the  eye,  on  the  one 
hand,  by  the  image  as  viewed  in  the  instrument,  and,  on  the  other  hand, 
by  the  object  as  seen  by  the  naked  eye  at  the  distance  of  distinct  vision. 
Denoting  this  ratio  by  the  symbol  W,  we  have  therefore: 

tan  6'      a    y' 
-^''  (449) 


Although  this  definition  of  the  subjective  magnifying  power  com- 
bines the  two  merits  of  simplicity  and  clearness,  it  is  open  to  objection 
on  account  of  the  fact  that  it  involves  essentially  the  magnitude  de- 
noted here  by  a,  the  so-called  "distance  of  distinct  vision",  which  has 
no  connection  with  the  instrument  itself  and  which  is  different  for 
different  individuals.  It  is  a  well-known  fact  of  experience  that  by 
virtue  of  its  power  of  accommodation  the  normal  eye  is  capable  of 
seeing  distinctly  at  almost  any  distance;  but  what  is  here  meant  by 
the  distance  of  distinct  vision  is  the  distance  from  the  eye  at  which 
an  observer  would  naturally  place  an  object  in  order  to  view  it  intently; 
which  in  the  case  of  a  normal  eye  is  usually  reckoned  as  about  25  cm. 
or  10  in.  Accordingly,  whereas  the  magnification  as  defined  by  the 
ratio  W  will  be  different  for  a  near-sighted  observer  for  whom  a  =  10 
cm.  and  for  a  far-sighted  observer  for  whom  a  =  50  cm.,  yet,  as  ABBE1 
has  pointed  out,  both  observers  looking  through  the  instrument  will, 
as  a  matter  of  fact,  view  the  image  of  the  same  object  under  the  same 
visual  angle  ;  so  that  whatever  difference  there  may  be  in  the  magnifi- 
cation is  to  be  found,  not  in  the  instrument  itself,  but  in  the  different 
organs  of  sight  that  are  employed  in  conjunction  with  the  apparatus. 

Eliminating  the  angle  t\  which  has  nothing  to  do  with  the  optical 
instrument,  we  may  write  the  formula  for  W  in  the  following  form: 


=  a-7,  •          (450) 

whereby  the  magnifying  power  W  is  expressed  now  as  the  product  of 
two  factors,  viz.,  the  factor  a,  which  depends  entirely  on  the  eye  of  the 

1  E.  ABBE:  Note  on  the  Proper  Definition  of  the  Amplifying  Power  of  a  Lens  or  a 
Lens-system:  Journ.  Roy.  Micr.  Soc.,  (2),  iv  (1884),  348-351.  See  German  translation 
in  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904),  445-449. 

See  also  S.  CZAPSKI:  Theorie  der  optischen  Instrumente  nacb  ABBE  (Breslau,  1893), 
160-164. 


548  Geometrical  Optics,  Chapter  XIV.  [  §  369. 

observer,  and  the  factor 

V=^,  (450 

which,  notwithstanding  the  fact  that  the  distance  of  the  image  from 
the  eye  is  involved  in  the  definition  of  the  apparent  size  tan  0'  of  the 
image,  depends  essentially,  as  we  shall  show,  on  the  structure  of  the 
optical  system  alone. 
Since 


where  x'  =  E'M',  xr  =  E'M'  denote  the  abscissae,  with  respect  to  the 
secondary  focal  point  £',  of  the  points  M'  and  M'  respectively,  and 
where  e'  denotes  the  secondary  focal  length  of  the  optical  system,  we 
obtain  : 


Now  almost  without  exception  in  the  case  of  all  optical  instruments 
that  are  employed  subjectively  in  conjunction  with  the  eye,  no  matter 
how  the  image  may  be  focussed  by  the  eye,  the  distance  x'  is  so  small 
in  comparison  with  the  distance  £  that  the  fraction  xf  /£'  is  practically 
negligible.  Under  these  circumstances  we  may  write  therefore  : 

tan  6'       i 
F=  -  -  =  -,  approximately;  (453) 

and  in  the  special  case  when  the  plane  or'  of  the  exit-pupil  coincides 
with  the  secondary  focal  plane  (x'  =  o)  and  the  eye  is  situated  at 
the  secondary  focal  point  £',  the  formula^  =  I  /e'  will  be  strictly  true. 
Accordingly,  as  above  stated,  the  magnitude  denoted  by  V  depends 
solely  on  the  structure  of  the  optical  instrument  provided  it  is  to  be 
used  subjectively. 

According  to  ABBE,  this  magnitude  V  defined  as  the  ratio  of  the 
visual  angle  subtended  at  the  eye  by  the  image  viewed  through  the  instru- 
ment to  the  corresponding  linear  dimension  of  the  object  is  therefore  a 
proper  measure  of  the  characteristic  or  intrinsic  magnifying  power  of 
an  optical  system  on  the  order  of  the  microscope.  For  every  such 
system  it  has  a  perfectly  definite  value,  viz.,  i/e',  and  thus  is  entirely 
independent  of  all  the  more  or  less  accidental  circumstances  that  may 
affect  the  magnification,  such  as  the  distance  from  the  image  of  the 
observer's  eye,  the  distance  from  the  focal  plane  of  the  exit-pupil,  etc. 


§  370.]  The  Aperture  and  the  Field  of  View.  549 

ABBE'S  definition  V  of  the  Subjective  Magnifying  Power  is  obtained 
from  the  ordinary  definition  W  by  merely  dividing  W  by  the  distance 
a  of  distinct  vision  of  the  observer;  thus, 

W 
V=~>  (454) 

Since  W  is  proportional  to  a,  the  popular  use  of  the  term  ''magnifying 
power",  which  corresponds  to  the  magnitude  W,  expresses  the  fact 
that  the  advantage  gained  by  the  use  of  an  optical  instrument  is 
proportional  to  the  observer's  distance  of  distinct  vision  and  is  there- 
fore greater  for  a  far-sighted  than  for  a  near-sighted  observer.  From 
the  scientific  point  of  view,  ABBE'S  definition  V  is  far  superior,  inas- 
much as  V  is  a  constant  of  the  instrument  itself.  The  subjective 
magnifying  power  V  in  the  case  of  an  instrument  on  the  order  of  the 
microscope  is  seen  to  be  completely  analogous  to  the  objective  magni- 
fying power  y' /tan  6  in  the  case  of  the  image  of  an  infinitely  distant  ob- 
ject formed  by  an  optical  instrument  on  the  order  of  the  photographic 
objective  or  the  objective  of  the  telescope. 

ART.    117.     THE   FIELD    OF   VIEW. 

370.     Entrance-Port  and  Exit-Port. 

The  limiting  of  the  bundles  of  rays  that  are  permitted  to  traverse 
the  optical  system  is  not  the  only  duty  performed  by  the  stops  and 
lens-fastenings;  but  these  serve  also  to  define  the  extent  of  the  object 
that  is  reproduced  in  the  image.  For  the  sake  of  simplicity,  let  us 
assume  for  the  present  that  the  aperture-stop  at  0  is  infinitely  small,  so 
that  the  pupil-openings  at  M  and  M'  (Fig.  162)  are  reduced  to  mere 
points  (9  =  6'  =  o,  p  =  p'  =  o).  In  this  case  the  chief  ray  of  a 
bundle  will  be  the  only  effective  ray,  and  the  bundle  of  chief  rays  will 
constitute  therefore  the  entire  system  of  effective  rays. 

In  order  now  to  ascertain  which  one  of  the  stops  present  is  the  one 
that  determines  the  expanse  of  object  that  will  be  depicted,  we  con- 
struct, as  before  (§  361),  the  image  of  each  stop  formed  by  that  part 
of  the  optical  system  which  is  in  front  of  it.  That  one  whose  image 
thus  constructed  subtends  at  the  centre  M  of  the  entrance-pupil  the 
smallest  angle  is  the  stop  that  limits  the  field  of  view  of  the  object.  In 
the  diagram  this  stop-image  is  represented  as  situated  with  its  centre 
on  the  optical  axis  at  the  point  designated  by  5.  The  cone  of  chief 
object-rays  whose  transversal  cross-section  at  5  coincides  with  this 
stop-image  divides  the  transversal  object-plane  <r  into  two  regions,  an 


550 


Geometrical  Optics,  Chapter  XIV. 


[§370. 


inner  circular  space  comprising  the  so-called  field  of  view  of  the  object 
and  an  outer  region  containing  points  so  situated  that  no  rays  emanat- 
ing from  them  can  go  through  the  instrument.  The  actual  stop  that 
is  thus  responsible  for  the  limiting  of  the  field  of  view  of  the  object  may 
be  called  the  field-stop;  as  seen  from  the  centre  0  of  the  aperture-stop 


06/ect 


FIG.  162. 


ENTRANCE-PORT  AND  EXIT-PORT  OF  OPTICAL  SYSTEM,  xx?  represents  the  optical  axis  of  the 
system,  which  latter  is  not  shown  in  the  diagram.  The  Aperture-Stop  is  here  assumed  to  be  a  pin- 
hole  opening  at  O ;  so  that  the  Entrance-Pupil  and  Exit-Pupil  are  likewise  contracted  into  mere 
points  at  M,  M '.  respectively.  In  the  Object-Space  are  shown  two  stop-images,  of  which  the  one 
whose  centre  is  at  the  point  marked  6"  subtends  at  M  the  smaller  angle.  This  stop-image  is  the 
Entrance- Port.  The  Exit-Port  with  its  centre  at  S'  subtends  at  M '  an  angle  smaller  than  that  sub- 
tended there  by  any  other  stop-image  on  the  image-side. 

0,  Z  S'M'T'  =  0'. 


MS-- 


ST=g, 


it  is  the  material  stop  or  lens-fastening  that  subtends  the  smallest 
visual  angle.  The  image  of  the  field-stop  formed  by  the  part  of  the 
optical  system  that  lies  in  front  of  it  may  be  appropriately  called 
the  Entrance-  Port.  Its  radius  ST  (reckoned  plus  or  minus  according 
as  the  circumference-point  T  is  above  or  below  the  axis)  will  be  denoted 
by  q.  Finally,  the  angle  SMT  =  0  subtended  at  the  centre  M  of 
the  entrance-pupil  by  the  radius  of  the  entrance-port,  defined  more 
precisely  by  the  relation 


tan©  = 


MS 


(455) 


where  c  =  MS,  is  called  the  angular  measure  (or  the  semi-angular 
diameter)  of  the  field  of  view  of  the  object. 

Analogously,  the  image  of  the  field-stop  produced  by  the  part  of 
the  optical  system  L  that  lies  on  the  other  side  of  it  (which  will  ob- 
viously be  identical  also  with  the  image  of  the  entrance-port  produced 
by  the  action  of  the  entire  compound  system  L)  will  define  likewise 
the  extent  of  the  image  and  may  be  called  the  Exit- Port.1  Thus, 

1  The  names  "  Entrance-Port  "  and  "  Exit-Port  "  introduced  here  were  suggested  by 
the  corresponding  terms  Eintrittsluke  and  AustriUsluke  used  by  VON  ROHR  in  his  treatise 


§  372.]  The  Aperture  and  the  Field  of  View.  551 

also,  the  angle  S'M'T'  =  0',  where  5',  T  designate  the  positions  of 
the  points  conjugate,  with  respect  to  the  entire  system,  to  the  points 
designated  above  by  5,  T,  respectively,  is  the  angular  measure  (or  the 
semi-angular  diameter)  of  the  field  of  view  of  the  image. 

It  is  possible,  of  course,  that  an  optical  system  may  have  two  or 
more  entrance-ports.  An  obvious  illustration  is  suggested  by  the 
familiar  type  of  photographic  double-objective  in  which  the  two  parts 
of  the  system  are  symmetrical  with  respect  to  the  aperture-stop  in 
the  middle  (as  in  the  case  of  the  "Aplanats"),  so  that  the  rims  of  the 
two  lens-systems  subtend  equal  angles  at  the  centre  0  of  the  aperture- 
stop;  and  hence,  since  the  rim  of  the  front  component  and  the  image 
of  the  rim  of  the  hinder  component  produced  by  the  front  component 
subtend  equal  angles  at  the  centre  M  of  the  entrance-pupil,  either  of 
these  two  may  be  regarded  as  the  entrance-port.  This  fact  will  be 
found  to  possess  a  certain  importance  in  the  case  of  an  optical  system 
of  finite  aperture,  as  we  shall  have  occasion  to  see  (§  383). 

371.  In  the  special  case  when  the  extent  of  the  object  MQ  is  so 
small  that  the  angle  subtended  at  the  centre  M  of  the  entrance-pupil 
is  smaller  than  the  angle  subtended  at  the  same  point  by  the  entrance- 
port  (that  is,  /.MMQ  <  Z.  SMT),  the  field  of  view  is  limited  by  the 
object  itself.     In  any  case  if  we  designate  by  Q  the  object-point  in 
the  transversal  plane  a  that  is  farthest  from  the  axis,  the  angular 
measure  of  the  field  of  view  of  the  object  is  Z.MMQ  —  0,  where  0 
denotes  always  the  slope-angle  of  the  outermost  ray  of  the  bundle  of 
chief  rays  in  the  Object-Space.     If  we  put  M  Q  =  y,  MM  =  £,  we 
can  write :  . 

tan0=£.  (456) 

ART.  118.     PROJECTION-SYSTEMS  WITH  INFINITELY   NARROW  APERTURE 

(6  =  0). 

372.  Focus-Plane  and  Screen-Plane.     According  to  the  geometri- 
cal theory  of  collinear  correspondence,  the  image  of  a  3 -dimensional 
object   is  itself    3 -dimensional;  but  by  the  image  produced  by  an 
optical  instrument  is  usually  meant  not  this  geometrical  image-relief 
in  space,  but  almost  without  exception  the  projection  thereof  on  some 
specified  surface,  such  as  the  retina  of  the  eye  itself  in  the  case  of  the 
class  of  optical  instruments  that  are  used  subjectively  in  conjunction 
with  the  eye,  or  such  as  a  screen  or  sensitive  photographic  plate  in 

on  Die  Strahlenbegrenzung  in  oplischen  Systemen.  (See  Die  Theorie  der  optischen  Instru- 
mente,  Bd.  I  (Berlin,  1904),  edited  by  M.  VON  ROHR:  Chapter  IX,  466-507.) 


552 


Geometrical  Optics,  Chapter  XIV. 


§372. 


that  other  large  class  of  optical  instruments,  the  so-called  projection- 
systems.  Only  at  such  points  of  this  selected  surface  as  are  conju- 
gate to  actual  points  of  the  object  will  the  definition  of  the  projected 
image  be  sharp;  these  points  being  the  vertices  of  bundles  of  rays 
which  emanated  originally  from  the  corresponding  points  of  the  object. 
But  since  the  image-points  conjugate  to  all  the  other  points  of  the 
object  will  lie  on  one  side  or  the  other  of  this  projection-surface,  these 
object-points  will  be  represented  on  this  surface  not  by  points  at  all, 


FIG.  163. 

PROJECTION-SYSTEM  WITH  INFINITELY  NARROW  APERTURE.  The  transversal  planes  <r,  a'  con- 
jugate to  each  other  with  respect  to  the  convex  lens  L  represent  here  the  Focus-Plane  and  the 
Screen-Plane,  respectively.  The  centres  of  the  infinitely  narrow  pupils  are  at  M ,  M ',  and  the  only 
effective  ray  of  each  bundle  is  the  chief  ray.  If  the  numerals  1,2,3,4  designate  the  positions  of 
certain  points  of  the  object,  and  if  1',  2',  3',  4'  designate  the  positions  of  the  corresponding  points 
of  the  image-relief,  the  Roman  numerals  without  primes  show  the  positions  in  the  Focus-Plane  of 
the  corresponding  vicarious  or  projected  object-points,  and  the  Roman  numerals  with  primes 
show  the  positions  in  the  Screen-Plane  of  the  corresponding  projected  points  of  the  image. 

but  by  the  sections  that  are  cut  out  of  the  corresponding  bundles  of 
image-rays  by  the  projection-surface.  Thus,  there  will  be  impressed 
on  the  surface  (retina  of  the  eye  or  screen)  a  certain  approximate 
effect  or  vicarious  image,  so  to  speak,  which  represents  the  relief- 
image  and  enables  us  to  form  a  more  or  less  correct  conception  of  it. 
In  the  case  of  the  photographic  objective,  which  may  be  considered 
as  a  typical  example  of  the  projection-system,  the  image  is  cast  upon  a 
plane  surface  placed  at  right  angles  to  the  optical  axis,  and  the  instrument 
is  focussed  on  a  definite  axial  object-point  M  by  adjusting  the  ground- 
glass  screen  so  that  its  front  surface  coincides  with  the  transversal 
plane  a'  conjugate  to  the  transversal  plane  a  of  the  axial  object-point 
M .  In  the  present  discussion  it  will  be  convenient  to  distinguish  the 
pair  of  conjugate  transversal  planes  a  and  a'  as  the  Focus- Plane  and 


§  373.]  The  Aperture  and  the  Field  of  View.  553 

the  Screen- Plane,  respectively.  In  case  the  aperture  is  infinitely  nar- 
row, as  is  here  assumed,  the  chief  ray  is  the  only  ray  of  the  bundle 
that  is  effective;  and  the  figure  in  the  focus-plane  corresponding  to 
that  which  is  actually  visible  on  the  screen-plane  may  be  constructed 
point  by  point  by  tracing  backwards  the  path  of  each  chief  ray  from 
the  point  Pr  where  it  crosses  the  screen-plane  to  the  point  P  where 
the  corresponding  ray  in  the  Object-Space  crosses  the  focus-plane. 
Practically,  this  process  amounts  simply  to  projecting  all  the  points 
of  the  object  from  the  centre  M  of  the  entrance-pupil  on  to  the  chosen 
focus-plane;  and  this  projection-figure,  which  may  be  called  the  "pro- 
jected object",  is  the  object  that  is  in  reality  reproduced  in  the  "pro- 
jected image"  in  the  screen-plane;  which  latter  may  also  be  constructed 
in  the  same  way  by  projecting  all  the  points  of  the  relief-image  from 
the  centre  M'  of  the  exit-pupil  on  to  the  screen-plane,  as  shown  in 
Fig.  163. 

373.    Perspective-Elongation. 

If  MQ  (Fig.  164)  is  the  projection  from  M  on  to  the  focus-plane  a 
of  an  object-line  NR  perpendicular  to  the  optical  axis  at  N,  we  have : 


MX 


PERSPECTIVE  ELONGATION  OF  THE  OBJECT.    Q  is  the  projection  from  centre  M  of  the  Entrance- 
Pupil  of  the  object-point  JR  on  to  the  Focus-Plane  <r. 


MM  MM 


or,  since  MN  is  usually  small  in  comparison  with  MM, 

MQ  MN 

~-  '  +       '  aPProximatelv- 


The  difference  MQ  —  NR  is  the  measure  of  the  perspective  elonga- 
tion of  the  object  NR,  and  the  ratio 

MQ  -  NR       MN 
NR          ~~  NM 

is  called  the  relative  perspective  elongation  of  the  object  NR. 


554 


Geometrical  Optics,  Chapter  XIV. 


[§374. 


CORRECT  DISTANCE  FOR  VIEWING  A  PHOTOGRAPH. 
Herey  -MCf  is  the  image  on  the  ground-glass  screen 
^  of  the  object  NR.  If  the  eye  is  placed  at  M ,  the  photo- 
graph  or  copy  placed  in  the  position  PK=  —y1  will  have 
the  same  perspective  as  the  object. 


374.     Correct  Distance  of  Viewing  a  Photograph. 
Viewed  from  the  centre  M  of  the  entrance-pupil,  the  object  NR  and 
the  projected  object  M Q  have  the  same  apparent  size ;  and  this  is  the 
case  no  matter  where  the  focus-plane  cr  is  situated.     In  order  to  get  the 

correct  impression,  the 
projected  image  M'Q'  = 
Y-MQ,  where  F  denotes 
the  value  of  the  magni- 
fication-ratio for  the  pair 
of  conjugate  transversal 
planes  cr,  cr',  should  be 
inspected  under  the  same 
visual  angle  6  that  is  sub- 
tended by  NR  or  M  Q  at 
the  centre  M  of  the  en- 
trance-] 

K/ 

is  the  image  projected  by 
a  photographic  objective  on  the  ground-glass  screen  of  the  object  NR, 
the  photograph  PK  =  Q'M'  =  —  y'  should  be  viewed  at  such  a  dis- 
tance d  =  PM  that  it  subtends  at  M  the  same  angle  0  as  the  object 
subtends  there.  From  the  figure,  with  the  aid  of  the  image-equations, 
it  can  readily  be  shown  that  the  correct  distance  of  viewing  a  photo- 
graph is: 

d-  =  e'(YZ-  i), 

where  e'(=  —  f)  denotes  the  secondary  focal  length  of  the  photo- 
graphic objective,  Z  denotes  the  angular  magnification  at  the  pupil- 
centres  Af,  M ',  and  Y  denotes  the  linear  magnification  with  respect 
to  the  given  focus-plane  cr.  Of  course,  in  using  this  formula,  attention 
must  be  paid  to  the  signs  of  the  magnitudes  represented  by  the  symbols 
therein.1  In  the  case  of  a  landscape-lens,  Y  =  o,  and  then  d  =  —  e'; 
which  means  that  the  photograph  of  a  landscape  should  be  viewed 
at  a  distance  equal  to  the  focal  length  of  the  lens.  Ordinarily,  we 
have  Z  equal  very  nearly  to  unity,  and  in  such  cases  (6  =  9')  the 
photograph  should  be  viewed  at  a  distance  d  =  M'M'  =  £'. 


1See  M.  VON  ROHR:  Theorie  und  Geschichte  des  photographischen  Objektivs  (Berlin, 
1899).  16. 


§  375.]  The  Aperture  and  the  Field  of  View.  555 

ART.    119.     OPTICAL   SYSTEMS   WITH   FINITE   APERTURE. 

375.  Projected  Object  and  Projected  Image  in  the  case  of  Pro- 
jection-Systems of  Finite  Aperture. 

So  long  as  the  aperture  of  the  system  was  infinitely  narrow,  we  had 
to  consider  merely  the  procedures  of  the  chief  rays;  but  advancing 
now  to  the  study  of  optical  projection-systems  of  finite  aperture,  we 
must  take  account  of  other  rays  besides  just  those  that  in  the  Object- 
Space  are  directed  towards  the  centre  of  the  entrance-pupil.  Every 
point  of  the  object  is  the  vertex  of  a  cone  of  rays  whose  paths  lie  along 
straight  lines  which,  produced  if  necessary,  must  first  of  all  go  through 
points  in  the  transversal  plane  <7  contained  within  the  circular  opening 
of  the  entrance-pupil.  Some  of  the  rays  of  such  a  bundle,  possibly 
all  of  them,  may  be  intercepted  at  the  entrance-port,  and  in  this  event 
only  a  portion  of  the  bundle  at  most  will  be  effective.  To  each  cone 
of  rays  in  the  Object-Space  corresponds  also  a  cone  of  rays  in  the 
Image-Space,  whose  paths  likewise  lie  along  straight  lines  which,  pro- 
duced if  necessary,  must  pass  through  points  in  the  transversal  plane 
<r'  comprised  within  the  circular  opening  of  the  exit-pupil;  and  to  an 
incomplete  cone  of  object-rays  corresponds,  of  course,  an  incomplete 
cone  of  image-rays.  The  relief-image  of  a  3 -dimensional  object  is  the 
configuration  of  image-points  which  are  at  the  vertices  of  all  these 
cones  or  partial  cones  of  image-rays.  Some  of  these  vertices  may  fall 
in  the  transversal  screen-plane  cr';  and  these  will  be  the  image-points 
corresponding  to  such  of  the  points  of  the  object  as  lie  in  the  trans- 
versal focus-plane  a.  But  all  the  other  points  of  the  object,  which 
lie  to  one  side  or  other  of  the  focus-plane,  will  be  represented  in  the 
projected  image  on  the  screen-plane,  not  by  points  at  all,  but  by  the 
circular  discs  or  patches — so-called  "diffusion-circles"  or  ' 'blur-circles" 
(see  §  365)— which  are  the  sections  of  the  cones  of  image-rays  made 
by  the  screen-plane.  In  the  case  of  an  incomplete  cone  of  image-rays, 
the  image  of  the  corresponding  object-point  will  be  represented  on  the 
screen-plane  by  only  a  piece  of  a  blur-circle.  These  ideas  will  be  made 
clear  by  the  consideration  of  the  diagram  (Fig.  166)  which  represents 
a  meridian  section  of  an  optical  system  consisting  of  an  infinitely  thin 
convex  lens  UT  with  a  front  stop  CD  with  its  centre  on  the  optical 
axis  at  M .  In  this  illustration  the  rim  of  the  lens  is  the  circumference 
of  both  the  entrance-port  and  the  exit-port. 

The  real  object  corresponding  to  the  above-described  projected 
image  in  the  screen-plane  a'  is  the  figure  in  the  focus-plane  a  obtained 
by  projecting  the  entrance-pupil  on  to  this  plane  from  each  point  of 
the  actual  object.  In  the  case  of  those  object-points  so  situated  that, 


556 


Geometrical  Optics,  Chapter  XIV. 


[  §  376. 


on  account  of  the  limited  opening  of  the  entrance-port,  they  can 
utilize  only  a  part  of  the  area  of  the  entrance-pupil,  we  must  project 
on  to  the  focus-plane  only  the  part  of  the  entrance-pupil  that  is  util- 
ized. The  centres  of  these  circular  discs  and  disc-portions  which  are 
the  sections  of  the  bundles  of  effective  rays  made  by  the  focus-plane 
a  and  the  screen-plane  a'  in  the  Object-Space  and  Image-Space,  re- 
spectively, are  at  the  points  where  the  chief  rays  cross  these  planes. 
This  last  statement  suggests  also,  that  in  regard  to  this  vicarious 
object-figure  in  the  focus-plane  a,  there  is  an  important  difference  to 
be  remarked  between  the  case  of  a  point  inside  of  one  of  these  object- 


PROJECTED  OBJECT  AND  IMAGE  IN  PROJECTION-SYSTEM  OF  FINITE  APERTURE.  The  Entrance- 
Pupil  CD  is  projected  from  the  object-point  R  on  to  the  Focus-Plane  <r  in  the  blur-circle  with  centre 
at  Q\  and.  similarly,  the  Exit-Pupil  C'L/  is  projected  from  the  image-point  R'  on  to  the  Screen- 
Plane  a'  in  the  blur-circle  with  centre  at  the  point  Q*  conjugate  to  Q. 

side  blur-circles  and  the  case  of  an  ordinary  object-point  lying  in  the 
focus-plane;  for  whereas  the  latter  emits  rays  in  all  directions,  the 
former  is  to  be  regarded  as  sending  out  only  one  single  ray  coinciding 
with  the  actual  object-ray  which  crosses  the  focus-plane  at  this  point. 

If  the  aperture  of  the  optical  system  is  not  only  finite  but  rela- 
tively large,  the  transversal  planes  a,  a'  must  be  a  pair  of  aplanatic 
planes  in  order  that  there  may  be  a  point-to-point  correspondence 
between  the  focus-plane  and  the  screen-plane;  and  when  this  is  the 
case,  the  image  of  an  object-point  which  lies  outside  the  focus-plane 
will  not  be  a  point,  since  the  so-called  HERSCHEL-Condition  (cf.  §  324) 
is  incompatible  with  the  Sine-Condition.  Under  such  circumstances, 
where,  in  general,  the  bundles  of  image-rays  are  no  longer  homocentric, 
it  is  particularly  advantageous  to  represent  the  image  of  a  3 -dimen- 
sional object  by  means  of  its  projected  image  on  the  screen-plane. 

376.  The  centres  of  the  blur-circles  on  the  screen-plane  are  to  be 
regarded  as  the  positions  of  the  image-points;  and  since,  even  in  the 
extreme  case  just  mentioned  of  a  system  of  very  large  aperture,  these 


§  378.]  The  Aperture  and  the  Field  of  View.  557 

are  the  places  where  the  chief  image-rays  cross  this  plane,  the  perspect- 
ive is  exactly  the  same  here  as  for  the  case  of  a  system  of  infinitely 
narrow  aperture  (§373),  so  that  nothing  needs  to  be  added  to  what 
has  been  said  already  in  the  treatment  of  the  perspective  in  the  pre- 
ceding case. 

377.  Focus-Depth  of  Projection-System  of  Finite  Aperture. 
With  regard  to  the  distinctness  of  the  image  on  the  screen-plane, 

that  is  a  matter  that  will  depend  very  largely  on  the  acuteness  of 
vision  of  the  observer.  If  the  resolving  power  of  the  eye  were  ab- 
solutely perfect,  this  screen-image  composed  partly  of  image-points 
and  partly  of  blur-circles  and  pieces  of  such  circles  would  appear 
faulty  on  the  mere  ground  that  it  was  not  a  faithful  reproduction  of 
the  original.  But  the  resolving  power  of  the  eye  is  limited  (cf.  §  252), 
depending  on  a  variety  of  conditions,  both  physical  and  physiological. 
Under  average  conditions  the  human  eye  is  able  to  distinguish  as  sepa- 
rate and  distinct  two  points  whose  angular  distance  apart  varies  for 
different  individuals  between  the  limits  of  one  and  five  minutes  of  arc;1 
and  hence  the  blur-circles  in  the  projection-image  will  not  be  dis- 
tinguishable from  points  provided  their  angular  diameters  do  not  ex- 
ceed this  limiting  angular  measure  (e)  of  the  resolving  power  of  the  eye. 
Similarly,  also,  in  regard  to  the  projection-figure  of  the  object  on  the 
focus-plane,  in  order  that  this  may  appear  sharp  and  distinct  as  viewed 
by  an  eye  at  the  centre  M  of  the  entrance-pupil,  the  diameters  of  the 
blur-circles  must  subtend  at  M  angles  that  are  smaller  than  the  limiting 
angle  e.  Since  the  diameters  of  these  blur-circles  will  depend  on  the 
distances  of  the  actual  object-points  from  the  focus-plane,  the  question 
arises  how  far  from  this  plane  can  such  an  object-point  be  in  order  that 
its  image  in  the  screen-plane  shall  still  appear  to  be  a  point  and  not 
a  fleck  of  light.  This  distance,  as  we  shall  see,  will  be  different  ac- 
cording as  the  object-point  lies  on  one  side  or  the  other  of  the  focus- 
plane,  so  that  all  object-points  which  are  comprised  within  the  space 
between  two  determinate  transversal  planes  at  unequal  distances  from 
the  focus-plane  and  on  opposite  sides  of  it  will  be  reproduced  distinctly 
in  the  projection-image  in  the  screen-plane.  The  distance  between 
this  pair  of  transversal  planes,  called  the  Focus-Depth,  we  propose  now 
to  investigate. 

378.  Let  <2t  (Fig.  167)  designate  the  position  of  the  point  where 
the  chief  ray  R^M  of  the  object-point  Rl  crosses  the  focus-plane  o-, 
so  that  (^  is  therefore  the  centre  of  the  blur-circle  that  represents  J^ 

lSec,  for  example,  E.  ABBE:  Beschreibung  eines  neuen  stereoskopischen  Oculars: 
CARLS  Rep.  f.  Exp.-Phys.,  xvii  (1881),  197-224.  See  p.  219.  This  paper  will  be  found 
also  in  Gesammelte  Abhandlungen,  Bd.  I  (Jena,  1904).  244-272. 


558 


Geometrical  Optics,  Chapter  XIV. 


§378. 


in  the  projection-figure  on  the  focus-plane.  The  object-ray  Rfi  which, 
lying  in  the  meridian  plane  of  the  figure,  is  directed  towards  the  point 
D  of  the  circumference  of  the  entrance-pupil  determines  by  its  inter- 


FOCUS-DEPTH  OF  PROJECTION-SYSTEM  OF  FINITE  APERTURE.    Diagram  represents  upper  half 
of  meridian  section  of  Object-Space. 


Q\Qt  =  dy,    MM  =  £,    MN\ 


i,    MNZ  = 


p,     Z  MMD  =  ©. 


section  with  the  focus-plane  a  the  point  Q2  in  the  circumference  of 
the  above-mentioned  blur-circle.  If  JVX  designates  the  point  where 
the  optical  axis  crosses  the  transversal  plane  of  the  object-point  Rlt 
we  obtain  from  the  figure: 


which,  if  we  put 
QiQ2  = 


__MN,  MN, 

MD  ~  MNl  ~  MM  - 

y,    MM  =  f ,    MNl 


MD  =  p, 


may  be  written  as  follows: 


Hence,  since 


dy  _      % 
P       *  +  «&' 

p  =  -  £-tan0; 


(457) 


where  9  =  /.MMD  denotes  the  angular  measure  of  the  aperture  of 
the  system  (§363),  we  derive  the  following  expression  for  the  radius 
dy  of  the  blur-circle  on  the  focus-plane : 


dy=- 


^rtanG. 


(458) 


This  formula,  which  is  given  by  CZAPSKI/  shows  that  the  size  of  the 
blur-circle  depends  not  only  on  the  aperture-angle  6  and  the  distance 
8%!  of  the  object-point  from  the  focus-plane,  but  also  on  the  distance 
£  of  the  focus-plane  from  the  entrance-pupil.  Moreover,  the  size  of 

1  S.  CZAPSKI:  Die  Theorie  der  optischen  Instrumenie  nach  ABBE  (Breslau,  1893),  p.  170. 


§  378.]  The  Aperture  and  the  Field  of  View.  559 

the  blur-circle  does  not  depend  on  the  distance  of  the  object-point 
from  the  optical  axis,  so  that  all  object-points  in  the  same  transversal 
plane  will  be  represented  in  the  projection-figure  on  the  focus-plane 
by  blur-circles  of  equal  diameters.  Thus,  for  example,  the  blur-circle 
of  the  axial  object-point  Nl  is  equal  to  that  of  R^  in  the  figure 

MV  =  Q,Q2  =  dy. 

The  straight  lines  QJ)  and  Q%M  determine  by  their  intersection  a 
point  R2  on  the  opposite  side  of  the  focus-plane  from  jRlf  which,  re- 
garded as  an  object-point,  will  be  represented  in  the  projection-figure 
on  the  focus-plane  by  a  blur-circle  whose  centre  is  at  the  point  Q2 
and  whose  radius  Q2Q{  =  —  dy  has  the  same  absolute  magnitude  as 
that  of  the  object-point  Rlt  Thus,  on  either  side  of  the  focus-plane 
there  is  a  certain  transversal  plane  characterized  by  the  fact  that  all 
object-points  in  this  plane  will  be  projected  on  to  the  focus-plane  in 
blur-circles  all  of  the  same  prescribed  size.  If  we  put  MN2  =  5£2, 
where  N2  is  used  to  designate  the  point  where  the  optical  axis  crosses 
the  transversal  plane  of  the  object-point  R2,  we  obtain  from  the  figure, 
exactly  as  in  the  case  of  the  similar  formula  above : 

dy  _  5£2 

Hence,  also,  we  find  for  the  distances  from  the  focus-plane  of  this  pair 
of  transversal  planes: 


and,  accordingly,  we  see  also  that  the  two  transversal  planes  deter- 
mined by  these  formulae  are  at  unequal  distances  from  the  focus- 
plane,  and,  in  fact,  that  the  front  one  of  the  two  planes  (in  the  figure 
the  one  containing  the  object-point  R^  is  always  nearer  to  the  focus- 
plane  than  the  other  plane. 

Now  if  the  magnitude  dy  is  such  that 

dy  e 

--tan-. 

where  c  denotes  the  angular  measure  of  the  resolving  power  of  the 
eye  (in  the  figure  e/2  =  /.VMM),  the  blur-circle  on  the  focus-plane 
corresponding  to  an  object-point  lying  anywhere  in  the  space  com- 
prised between  the  pair  of  transversal  planes  belonging  to  i?t  and  R2 


560  Geometrical  Optics,  Chapter  XIV.  [  §  380. 

will  be  so  small  that  the  eye  placed  at  the  centre  M  of  the  entrance- 
pupil  could  not  distinguish  them  from  points;  so  that,  practically 
speaking,  all  object-points  lying  within  this  region  on  either  side  of 
the  focus-plane,  will  be  sharply  denned  in  the  projection-figure. 
The  distance 

=  N,M  +  MN2,  =  8£2  -  5{t 


between  the  pair  of  transversal  planes  determined  by  this  critical  value 
dy  =(e-£)/2  is,  as  was  stated  above,  the  Focus-Depth  of  the  project- 
ion-system for  a  given  position  of  the  focus-plane  er.  Thus,  we  find 
the  following  expression  for  the  Focus-Depth  : 


The  reciprocal  of  the  focus-depth  may  be  regarded  as  a  measure  of 
the  exactness  of  the  focus. 

379.    Lack  of  Detail  in  the  Image  due  to  the  Focus-Depth. 

As  to  the  detail  or  distinctness  of  the  image  projected  on  the  screen- 
plane,  this  is  a  question  that  involves  not  merely  the  absolute  sizes 
of  the  blur-circles  but  the  magnification-ratio  also.  Thus,  for  example, 
the  blur-circles  of  the  image  of  a  piece  of  hand-writing  which  is  magni- 
fied to  double  the  size  of  the  original  may  have  blur-circles  of  twice  as 
great  diameters  as  would  be  permissible  if  it  were  not  magnified  at  all  ; 
the  former  would  be  just  as  legible  or  distinct  as  the  latter.  Accord- 
ingly, as  a  measure  of  the  indistinctness  due  to  being  out  of  focus,  it 
has  been  proposed1  to  take  the  ratio  of  the  radius  dy'  of  the  blur- 
circle  in  the  screen-plane  cr'  corresponding  to  an  object-point  R  (see 
Fig.  165)  to  the  magnification  YS  =  M'Q'/NR.  Since  Y  =M'Q'/MQ, 
dy'=Y-dy  and  (§373) 

MQ=       { 
NR      £  +  5£' 

the  measure  of  the  indistinctness,  according  to  the  above  definition, 
is  given  as  follows: 

.  (461) 


380.  Focus-Depth  of  Optical  Systems  of  Finite  Aperture  used  in 
Conjunction  with  the  Eye. 

If  the  optical  system  is  to  be  employed,  not  for  the  purpose  of  casting 
objective  images  on  a  screen,  but  in  conjunction  with  the  eye  for  the 

See  S.  CZAPSKI:   Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  p.  171. 


§  381.]  The  Aperture  and  the  Field  of  View.  561 

reinforcement  of  vision,  and  if  the  pupil  of  the  passive  eye  is  supposed 
to  be  placed  at  the  exit-pupil  of  the  instrument,  the  image  is  presented 
to  the  eye  at  the  distance  M'M'  =  £'. 

The  absolute  linear  diameter  of  the  blur-circle  in  the  image-plane 
corresponding  to  a  non-focussed  point  is  : 

2dyf  =  2Y-dy=  -2Y-  j-^-  tan  6  =  -  2F-3£-tan  9,  approx., 

where  Y  denotes  the  lateral  magnification  of  the  aplanatic  pair  of 
axial  points  M,  Mr,  and  where,  in  obtaining  the  final  approximate 
expression,  the  distance  5£  is  supposed  to  be  small  as  compared  with  £, 
as  is  the  fact  with  an  optical  instrument  of  high  magnifying  power. 
If 

2dy'  _  6£ 

e  =  -£-  =  -  2F-  -j  -tan  0 

denotes  the  visual  angle  subtended  at  the  eye  by  the  blur-circle,  and 
if  we  recall  from  §  369  that 

F  =  tane_'_/  i     F 

y    V  r  "  .r  .  • 

we  find 
or 

.  .        (462) 

Thus,  if  y'fy  =  100,  and  if  the  absolute  value  of  £'  is  equal  to  the  con- 
ventional distance  of  distinct  vision,  viz.,  250  mm.,  so  that 


and  if  we  take  0  =  —  30°,  6  =  3'  =  0.00087  radian,  we  obtain  for  the 
Focus-Depth:  25£  =  0.0037  mm. 

381.    Accommodation-Depth. 

By  virtue  of  its  power  of  accommodation,  the  eye  can  be  focussed 
at  will  on  different  points  of  the  image-relief,  and  provided  these 
image-points  are  within  the  range  of  distinct  vision,  and  also  provided 
the  imagery  is  ideal,  the  different  parts  of  the  image  can  be  viewed  with 
perfect  exactness;  so  that,  owing  to  this  property  inherent  in  the  eye 
to  a  greater  or  less  degree  in  different  individuals,  a  certain  depth  of 
the  object  called  the  accommodation-depth  will  be  seen  distinctly  in 

37 


562  Geometrical  Optics,  Chapter  XIV.  [  §  381. 

its  image,  which  measured  along  the  axis  may  be  denoted  by  M^My 
The  depth  of  vision  is  extended  beyond  these  points  by  the  focus- 
depth  6£,  in  one  direction  from  Ml  and  the  focus-depth  S£2  in  the  other 
direction  from  M2,  since  within  these  extended  parts  the  blur-circles  are 
too  small  to  be  resolved  by  the  eye;  and  hence  the  Entire  Depth  of 
Vision  is  equal  to  the  sum  of  the  Accommodation-Depth  and  Focus- 
Depth,  viz.  =  M^  +  d£t  +  5£2.. 

If  the  eye  is  placed  at  the  exit-pupil  of  the  instrument,  whose  centre 
is  at  the  point  designated  by  M'  ,  and  if  the  positions  on  the  optical 
axis  in  the  Image-Space  of  the  "near-point"  and  "far-point"  of  the 
eye  of  the  observer  are  designated  by  M[  and  M'2J  respectively,  the 
range  of  distinct  vision  is  equal  to  the  piece  M(M2  of  the  optical  axis. 
The  points  designated  above  by  M^  and  M2  are  the  axial  object-points 
conjugate  to  M(  and  M'2,  respectively.  If  the  focal  points  of  the 
optical  system  are  designated  by  F  and  E't  and  if  we  put 

*!  =  FMlt    x2  =  FMV    x{  =  E'M(,    x2  =  E'M2, 

and,  finally,  if  the  focal  lengths  are  denoted  by  /  and  e',  then,  on  the 
assumption  of  collinear  correspondence,  we  have: 


and  hence: 

MtM,  =  S*  =  *2  -  *t 
where  dx'  =  AfiJf,.     If  (§  179) 


denote  the  magnification-ratios  of  the  two  pairs  of  conjugate  axial 
points,  and  if  we  introduce  also  the  relation  (§  193)  : 

n'f  +  ne'  =  o, 

where  n,  n'  denote  the  indices  of  refraction  of  the  first  and  last  media 
of  the  optical  system,  we  obtain: 

n       dx' 


If  M'  Mi  =  £i  and  M'  M?2  =  ^  denote  the  least  and  greatest  distances 
of  distinct  vision  of  the  eye,  then,  according  to  DONDERS,  the  magni- 
tude 

.      i        I       M(M2       8x' 

A=~fS=         ~  =      f  (463) 


§  382.]  The  Aperture  and  the  Field  of  View.  563 

is  the  rational  measure  of  the  power  of  accommodation  of  the  eye;1 
and  hence  we  obtain  the  following  expression  for  the  accommodation- 
depth  : 

~—~  ;  (464) 

and  if  Yl  and  Y2  are  not  much  different  from  each  other,  we  can 
replace  each  of  them  by  a  certain  mean  value  Y,  which,  to  be  perfectly 
accurate,  should  be  the  geometric  mean  between  Yl  and  F2;  and, 
similarly,  we  can  introduce  in  place  of  (£  and  £2  a  mean  value  £';  so 
that  the  final  form  of  the  expression  becomes: 


where  usually  £'  is  put  =  250  mm.,  the  conventional  distance  of  dis- 
tinct vision.  Thus,  for  example,  in  the  case  of  a  myopic  eye,  for 
which  £  =  150  mm.,  ^  =  3°°  mm->  so  tnat  A  =  1/300,  we  obtain  for 
a  magnification  of  Y  =  100  (assuming  n  =  n'  —  i): 


x  =  1/48  =  0.021 


mm. 


ABBE,2  who  has  investigated  this  subject  very  exhaustively,  espe- 
cially in  connection  with  the  microscope,  gives  several  tables  (which 
are  given  also  by  CzAPSKi3)  exhibiting  the  relations  between  the 
Focus-Depth  and  the  Accommodation-Depth  for  different  values  of 
the  magnification-ratio  F;  whereby  it  appears  that,  although  for  low 
magnifications  the  accommodation-depth  is  far  more  important  than 
the  focus-depth,  the  reverse  is  true  in  the  case  of  high  magnifications. 

ART.  120.     THE   FIELD   OF  VIEW  IN  THE   CASE    OF  PROJECTION-SYSTEMS 
OF    FINITE    APERTURE. 

382.    Case  of  a  Single  Entrance-Port. 

The  characteristic  effect  of  a  finite  aperture  in  dividing  the  field  of 
view  into  separate  regions  distinguished  by  the  different  magnitudes 
of  the  apertures  of  the  bundles  of  rays  that  have  their  vertices  at 

1  The  measure  of  the  power  of  accommodation  of  the  eye  is  the  strength  of  an  infinitely 
thin  lens,  placed  where  the  eye  is,  for  which  the  far- point  and  near- point  are  conjugate 
points. 

2  E.  ABBE:  Beschreibung  eines  neuen  stereoskopischen  Oculars:  CARLS  Rep.  f.  Exp.- 
Phys.,  xvii  (1881),  197-224:   also    Gesammelte  Abhandlungen,  Bd.  I    (Jena,  1904),  244- 
272.     Section  III  of  this  paper  treats  of  the  special  matters  here  referred  to.     See  also: 
E.  ABBE:  Journ.  Roy.  Micr.  Soc.  (2),  I  (1881),  687-689. 

3  S.  CZAPSKI:   Theorie  der  optischen  Inslrumente  nach  ABBE  (Breslau,  1893),  p.  173. 


564 


Geometrical  Optics,  Chapter  XIV. 


[  §  382. 


points  comprised  within  these  regions  was  remarked  by  J.  PETVZAL1 
in  the  case  of  a  photographic  double-objective  in  which  there  was  no 
material  diaphragm  other  than  the  lens-fastenings  themselves.  The 
investigation  of  this  effect  in  the  general  case  of  an  optical  projection- 
system  of  finite  aperture  will  be  different  according  as  the  field  of 
view  is  limited  by  one  or  by  two  ports;  and  hence  we  shall  treat,  first, 
the  simpler  case  of  an  optical  system  with  a  single  entrance-port. 

In  the  diagram  (Fig.  168)  the  plane  of  the  paper  represents  a  meri- 
dian section  in  the  Object-Space ;  so  that,  in  order  to  have  a  complete 
representation,  the  entire  figure  should  be  imagined  as  revolved 


FIG.  163. 

FIELD  OF  VIEW  OF  OBJECT  IN  CASE  OF  PROJECTION-SYSTEM  OF  FINITE  APERTURE  WITH  A 
SINGLE  ENTRANCE-PORT. 

£,    JUS=c,    MD  =  P,    ST=c.     L  SM  T=  Z  MM  Y=  0,    £  SHT=  £  MHX=  *. 


L  SGT= 

around  the  optical  axis  xx.  The  positions  on  the  axis  of  the  centres 
of  the  entrance-pupil  and  entrance-port  are  designated  by  M  and  5, 
respectively.  The  end-points,  on  the  same  side  of  the  axis,  of  the 
diameters,  in  the  meridian  plane  of  the  figure,  of  the  entrance-pupil 
and  entrance-port  are  designated  by  D  and  T,  respectively.  Finally, 
the  point  M  designates  the  point  where  the  optical  axis  crosses  the 
focus-plane  <r. 

The  straight  line  DT  joining  the  end-points,  on  the  same  side  of 

1  J.  PETZVAL:  Bericht  ueber  dioptrische  Untersuchungen:  Silzungsberichte  der  math.- 
naturw.  Cl.  der  kaiserl.  Akad.  der  Wissenschaften  (Wien),  xxvi  (1857),  33-90.     See  p.  57. 


1 382,]  The  Aperture  and  the  Field  of  View.  565 

the  axis,  of  the  diameters  of  the  entrance-pupil  and  entrance-port  meets 
the  optical  axis  at  the  point  designated  by  H  and  crosses  the  focus- 
plane  at  the  point  designated  byX.  The  region  of  the  field  of  view  of 
the  object  defined  by  the  circle  described  in  the  focus-plane  around  M 
as  centre  with  radius  equal  to  M. X  is  distinguished  by  the  fact  that  with- 
in this  circular  space  are  contained  all  the  points  of  the  focus-plane  a 
that  are  the  vertices  of  cones  that  have  the  entire  opening  of  the  en- 
trance-pupil as  common  base;  so  that  no  object-ray  emanating  from 
a  point  of  this  central  region  of  the  focus-plane  and  directed  towards 
a  point  of  the  circular  opening  of  the  entrance- pupil  will  be  intercepted. 

The  straight  line  MT  crosses  the  focus-plane  at  a  point  designated 
by  F,  which,  since  the  entrance-pupil,  in  consequence  of  its  definition 
(§  361),  must  subtend  at  M  a  smaller  angle  than  is  subtended  there  by 
the  entrance-port,  will  lie  always  on  the  same  side  of  the  optical  axis 
as  the  point  X  and  at  a  distance  MY  greater  than  MX.  The  annular 
region  of  the  field  of  view  comprised  between  the  circumferences  of 
the  two  concentric  circles  described  around  M  as  centre  with  radii 
equal  to  MX  and  MY  contains  all  points  which,  regarded  as  object- 
points,  are  in  a  position  to  utilize  one  half  or  more  of  the  total  aperture 
of  the  entrance-pupil.  Not  more  than  half  of  the  rays  of  a  bundle  of 
rays  emitted  from  an  object-point  in  this  annular  region  of  the  focus- 
plane  and  directed  towards  all  the  points  of  the  entrance-pupil  will 
be  intercepted,  and  in  general  less  than  half. 

Finally,  the  straight  line  joining  the  extremity  T  of  the  diameter 
of  the  entrance-port  with  the  opposite  extremity  of  the  diameter  of 
the  entrance-pupil  will  determine  by  its  intersection  with  the  focus- 
plane  a  a  third  point  Z,  also  on  the  same  side  of  the  axis  as  the  points 
X  and  F,  but  the  most  distant  one  of  the  three,  which  marks  the 
extreme  limit  on  that  side  of  the  axis  of  the  field  of  view.  More  than 
half  of  the  rays  emitted  by  an  object-point  lying  within  this  outside 
annular  space  of  the  focus-plane  that  are  directed  towards  all  the 
points  of  the  entrance-pupil  will  be  cut  off;  and  a  point  lying  in  the 
focus-plane  at  a  distance  from  the  axis  greater  than  M  Z  can  send 
through  the  system  no  ray  at  all. 

The  Z  MH  X  is  called  by  VON  RoHR1  the  vignette-angle.  Employing 
symbols  as  follows: 

£MHX  =  Z.SHT  =  M,    MM  =  £,    MS  =  c,    MD  =  p,    ST  =  g, 

we  obtain  from  the  figure: 

p-q  p-  MX 

tan  /*  =  -  ^~c~  =  -  -  -y-  -  ; 

1  M.  VON  ROHR:  Die  Theorie  der  optischen  Instrumente,  Bd.  I  (Berlin,  1904),  p  .485. 


566  Geometrical  Optics,  Chapter  XIV.  [  §  382. 

whence  also  we  find  the  following  expression  for  the  radius  of  the 
central  region  of  the  field  of  view: 

MX  =  p-^-t.  (466) 

The  abscissa  of  the  point  H  with  respect  to  the  centre  M  of  the  en- 
trance-pupil is: 

•          Jiyfrr  *  *  /j/c«\ 

•te^.-#^-  (46?) 

From  the  figure  also  we  obtain  the  following  relations: 

MXi_MD_    MH      MD  _  MD 
'"  MM  ~  MM  *  MH  ~  MM     MH 

=  -  tan  Z.MMD  -f  tan  Z MHX. 

The  Z.MMD  =  0  is  the  aperture-angle,  and  if  we  put  Z.MMX  —  x» 
the  result  just  obtained  may  be  written  as  follows: 

tan  x  =  tan  ^  —  tan  0 ;  (468) 

and  hence  the  tangent  of  the  angle  x  subtended  at  the  centre  M  of 
the  entrance-pupil  by  the  radius  MX  of  the  central  region  of  the  field 
of  view  is  equal  to  the  algebraic  difference  of  the  tangents  of  the 
vignette-angle  \L  and  the  aperture-angle  0.  In  terms  of  the  given 
linear  magnitudes,  we  can  write  also: 

tan  x  = —  +  7  -  (469) 

If  G  designates  the  position  of  the  point  where  the  straight  line  TZ 
crosses  the  optical  axis,  and  if  we  put  Z.SGT  =  Z.MGZ  =  X,  we 
obtain  from  the  figure  exactly  as  above : 

.    ;      ta^=p±^=p+^t 

C  s 

and  hence  for  the  radius  of  the  entire  field  of  view  we  find: 

MZ  =  — —  %-  p.  (470) 

The  abscissa  of  the  point  G  with  respect  to  the  centre  M  of  the  entrance- 
pupil  is: 


§  382.]  The  Aperture  and  the  Field  of  View.  567 

Moreover,  from  the  figure: 

MZ       MD     MG_MD      MD 
tan  ZMMZ  =  ^  ~  MM  '  MG  ~  MG  ~~  MM 

=  tan  Z  SGT  +  tan  ^MMD. 
If  we  put  Z.MMZ  =  ^,  this  result  may  be  written  as  follows: 

tan  \l/  =  tan  X  +  tan  ©.  (472) 

Hence,  the  tangent  of  the  angle  $  subtended  at  the  centre  M  by  the 
radius  MZ  of  the  entire  extent  of  the  field  of  view  is  equal  to  the 
algebraic  sum  of  the  tangent  of  the  angle  X  subtended  at  G  by  the  same 
radius  and  the  tangent  of  the  aperture-angle  9.  In  terms  of  the 
given  linear  magnitudes,  we  can  write  also : 

=  _P,P_±4.  (473) 


In  the  case  of  an  object-point  in  the  focus-plane  a  whose  chief  ray 
has  a  slope-angle  6  greater  than  the  angle  denoted  by  x,  a  part  of 
the  bundle  of  object-rays  will  be  intercepted  at  the  entrance-port. 
The  chief  rays  will  be  absent  from  the  bundles  of  effective  rays  that 
come  from  the  object-points  in  the  focus-plane  that  are  farther  from 
the  axis  than  the  point  designated  by  Y. 

Of  all  possible  object-rays  that  pass  unimpeded  through  the  centre 
M  of  the  entrance-pupil,  those  (such  as  YT)  that  graze  the  "rim"  of 
the  entrance-port  have  the  greatest  slopes,  viz. : 

AMMY  =  ASMT=  0, 

where  0,  in  the  case  of  an  infinitely  narrow  aperture,  is  the  angular 
measure  of  the  field  of  view  (§  370).  The  three  angles  at  M  denoted 
by  x»  0  and  \f/  define  the  limits  of  the  three  parts  of  the  field  of  view  of 
the  object. 

In  connection  with  the  case  of  an  optical  system  of  finite  aperture 
with  a  single  entrance-port,  one  point  remains  to  be  particularly  men- 
tioned, viz.,  with  respect  to  the  projection-figures  on  the  focus-plane 
of  object-points  that  lie  outside  this  plane.  If  the  slope-angle  9  of 
the  chief  ray  from  an  object-point  R  not  in  the  focus-plane  is  greater 
than  the  angle  %»  the  projection-figure  on  the  focus-plane  will  not  be 
a  circle  but  a  lune,  and  the  chief  ray  will  not  be  the  representative 
ray  of  the  bundle,  and  indeed,  if  /.MMR  >  0,  the  so-called  chief  ray 
will  be  absent  from  the  bundle  of  effective  rays  emitted  by  the  object- 


568 


Geometrical  Optics,  Chapter  XIV. 


[§383. 


point  R.  Hence,  also,  in  the  case  of  object-points  thus  situated,  it  is 
obviously  not  correct  to  consider  the  points  where  their  chief  rays 
cross  the  focus-plane  as  the  representative  points  of  their  projection- 
figures,  especially  since  the  former  may  not  even  lie  within  the  bound- 
aries of  the  corresponding  projection-figures  at  all. 

383.    Case  of  Two  Entrance-Ports. 

Proceeding  now  to  consider  the  case  of  an  optical  projection-system 
of  finite  aperture  with  two  entrance-ports,  we  may  regard  as  typical 
thereof  the  case  shown  in  the  diagram  (Fig.  169),  which  represents  a 


FIG.  169. 

FIELD  OP  VIEW  OF  OBJECT  IN  CASE  OF  PROJECTION-SYSTEM  OF  FINITE  APERTURE  WITH  Two 
ENTRANCE-PORTS. 


MD  =  p.    Si  T\ 


L  MH\X  -MI. 


=  8. 


meridian  section  of  the  Object-Space.  Let  Sv  S2  designate  the  centres 
and  7\,  T2  the  extremities  above  the  optical  axis  xx  of  the  diameters, 
in  the  meridian  plane  of  the  figure,  of  the  two  entrance-ports  ;  and  let 
I/i,  U2  designate  the  other  two  ends  of  these  diameters.  According 
to  our  previous  definitions,  the  centre  of  the  entrance-pupil  must  lie 
at  the  point  M  where  the  straight  line  T-JJ2  which  joins  a  pair  of 
opposite  ends  of  the  diameters  U^T^  U2T2  crosses  the  optical  axis. 
The  point  M  where  the  optical  axis  crosses  the  focus-plane  a  is  deter- 
mined by  the  intersection  of  the  straight  line  T2Tl  with  the  straight 
line  xx.  It  is  obvious  from  the  figure  that  the  points  M,  M  are  har- 
monically separated  by  the  points  Slt  S2,  so  that  we  have  the  relation: 


(474) 


§  384.]  The  Aperture  and  the  Field  of  View.  569 

If,  therefore,  the  positions  of  the  two  entrance-ports  with  reference 
to  the  entrance-pupil  are  given,  and  if  we  put: 

MSl  =  clt    MS2  =  c2, 
the  position  of  the  focus-plane  is  determined  by  the  relation: 


where  £  =  MM  denotes  the  abscissa  of  the  point  M  with  respect  to 
the  centre  M  of  the  entrance-pupil. 

Let  C  and  D  designate  the  lower  and  upper  ends,  respectively,  of 
the  diameter  of  the  entrance-pupil  which  lies  in  the  meridian  plane  of 
the  figure,  and  through  the  upper  ends  D  and  7\  of  the  diameters  CD 
and  Ul  Tl  draw  the  straight  line  D  T\  crossing  the  optical  axis  at  the 
point  designated  by  H^  and,  similarly,  through  the  lower  ends  of 
the  diameters  CD  and  U2T2  draw  a  straight  line  CU2  crossing  the 
optical  axis  at  the  point  designated  by  H2.  Let  X  designate  the  point 
of  intersection  of  the  straight  lines  DT^  and  CU2;  we  wish  to  show 
that  this  point  X  will  fall  in  the  focus-plane  a.  Suppose  it  does  not, 
and  that  we  draw  through  X  a  straight  line  parallel  to  CD  meeting 
the  optical  axis  in  a  point  not  marked  in  the  diagram  which  we  shall 
call  N,  and  meeting  the  straight  line  U-JM^  in  a  point  Y.  According 
to  this  construction,  it  is  plain  that  the  pair  of  points  TV,  M  will  be 
harmonically  separated  by  the  pair  of  points  H2,  H^  and  that  from 
the  point  X  the  harmonic  point-range  TV,  M,  H2,  Hl  will  be  projected 
on  to  the  straight  line  U^  in  the  harmonic  point-range  F,  M,  U2,  7^; 
which  latter  projected  on  to  the  optical  axis  from  the  infinitely  distant 
point  of  the  straight  line  CD  will  give  : 

(NMS2S,)  =  -  i. 

But  since,  as  a  matter  of  fact,  we  know  that 

(MMS2S,)  =  -  i, 

it  follows  that  the  point  designated  by  N  must  be  coincident  with  the 
point  Mj  and  hence  the  point  X  must  lie  in  the  focus-plane  cr,  as  shown 
in  the  figure.  Moreover, 

(MMH.H,)  =  -  i.  (475) 

384.  Any  object-point  lying  in  the  focus-plane  within  the  central 
region  defined  by  the  circle  described  around  M  as  centre  with  radius 


570  Geometrical  Optics,  Chapter  XIV.  [  §  384. 

equal  to  M  X  will  be  in  a  position  to  send  out  rays  that  will  go  through 
every  point  of  the  opening  of  the  entrance-pupil;  whereas  any  point 
in  the  focus-plane  at  a  greater  distance  from  M  than  X  will  be  in  a 
position  to  send  out  rays  that  will  go  through  some,  but  not  all,  of  the 
points  of  the  entrance-pupil,  provided  its  distance  from  M  does  not 
exceed  the  distance  M  F;  in  which  latter  case  it  cannot  send  any  rays 
through  the  optical  system. 
If  we  put 

Z  S.H.T,  =  Z  MH,X  =  Ml,     Z  S2H2T2  =  Z  MH^X  =  »2, 
MD  =  p,     5^  =  2!,     S2T2=q2, 

we  obtain  from  the  figure  : 

gi-/>  P        MX  -p 

-I—.-—  ---  _  , 

q2  -  p         p         MX  +  p 
tanM2=        —  -=  —  =      —  --; 

whence  also  we  find  for  the  radius  of  the  central  region  of  the  field 
of  view: 

MX  =  P  +  ^-P-  £--#•-  ^^  &  (476) 

Cl  C'2 

and  for  the  abscissae,  with  respect  to  M,  of  the  points  Hlt  H2: 

M>H=-^P'  MH*=-A          (477) 

Likewise  from  the  figure  we  obtain  also  the  following  relations: 

MX  MD       MD       MD       MD 

tan  Z  M.M.X.  =  ,«,,..  =  —  njTrT  ~\~  **-**  ==  **™  —  ****•  5 
MM          MHl      MM     MH2      MM 

and  hence  if  /.MMD  =  6,  LMMX  =  x,  we  have  here: 

tan  %  =  tan  ^  —  tan  0  =  tan  /*2  +  tan  6.  (47$) 

If  we  put  Z.MMY  =  0,  we  obtain  evidently  also: 


and  for  the  radius  of  the  entire  field  of  view: 

'*-«-  (479) 


Thus,  we  see  that,  whereas  the  position  of  the  point  F  is  entirely 
independent  of  the  diameter  of  the  entrance-pupil,  this  is  not  true 


§  387.]  Intensity  of  Illumination.  571 

with  regard  to  the  position  of  the  point  X;  for  the  greater  this  diameter 
is,  the  nearer^"  will  be  to  the  axial  point  M\  and  in  the  limiting  case 
when  the  end-point  D  of  the  diameter  of  the  entrance-pupil  lies  in  the 
straight  line  MT^T^  the  point X  will  coincide  with  M. 

385.  By  placing  in  the  focus-plane  a  circular  diaphragm  with  its 
centre  at  M  and  with  an  opening  of  radius  equal  to  MX,  all  of  the 
field  of  view  outside  the  central  part  will  be  screened  off ;  and  then,  pro- 
vided the  object  lies  wholly  in  the  focus-plane,  all  the  points  of  the 
object  will  send  through  the  system  cones  of  rays  that  fill  completely 
the  opening  of  the  entrance-pupil.     The  same  result  will  be  obtained 
by  placing  in  the  screen-plane  or  image-plane  a'  a  diaphragm  with  its 
centre  at  the  point  M'  conjugate  to  M  and  with  an  opening  of  radius 
M 'X'  =  F  •  MX,  where  F  denotes   the    magnification-ratio  of  the 
pair  of  conjugate  transversal  planes  <r  and  cr'.     Thus,  for  example, 
in  the  case  of  the  astronomical  telescope,  a  diaphragm  of  this  kind  is 
placed  in  the  focal  plane  of  the  objective.     This  simple  method  is 
applicable  to  all  cases  in  which  the  depth  of  the  object  is  negligible, 
especially  when  the  object-distance  is  prescribed  and  the  points  M,  Mf 
are  the  pair  of  aplanatic  points  of  the  optical  system  (§  279).     But  if 
the  points  of  the  object  are  situated  at  finite  distances  from  the  focus- 
plane,  a  stop  such  as  above  described  will  not  avail  for  this  purpose. 

386.  Consider  an  object-point  in  the  plane  of  the  figure  above  the 
optical  axis;  if  it  lies  to  the  right  of  the  focus-plane  within  the  angle 
MH2X  =  ju2  or  on  the  other  side  of  this  plane  within  the  angle MH^X  =  /tlf 
it  will  be  in  a  position  to  send  through  the  optical  system  a  cone  of  rays 
completely  filling  the  opening  of  the  entrance-pupil.     If  the  object- 
point  lies  within  the  angle  subtended  at  X  by  the  diameter  CD  of  the 
entrance-pupil,  some  of  the  rays  of  the  cone  which  has  the  opening  of 
the  entrance-pupil  for  its  base  will  be  intercepted  at  the  entrance- 
port  S2  if  the  vertex  of  the  cone  lies  to  the  right  of  the  focus-plane,  and 
at  the  entrance-port  5L  if  the  vertex  of  the  cone  lies  on  the  other  side 
of  the  focus-plane.     And,  finally,  if  the  object-point  lies  to  the  right 
of  the  focus-plane  and  outside  the  angle  /*p  or  on  the  other  side  of  the 
focus-plane  outside  the  angle  /*2,  all  the  rays  will  be  intercepted. 

INTENSITY   OF   ILLUMINATION   AND    BRIGHTNESS. 
ART.  121.     FUNDAMENTAL  LAWS  OF  RADIATION. 

387.  Radiation  of  Point-Source. 

Regarding  the  light-rays  as  the  routes  of  propagation  of  light-energy, 
we  may  call  a  bundle  of  rays  a  "tube  of  light";1  and  it  is  assumed 

1  See  P.  DRUDE:  Lehrbuch  der  Optik  (Leipzig,  1900),  p.  72.     See  also  P.  G.  TAIT:  Light 
(Edinburgh,  1889),  Chapter  V. 


572  Geometrical  Optics,  Chapter  XIV.  [  §  387. 

in  the  theory  of  radiation  that  with  a  steady  source  of  light  equal 
quantities  of  light-energy  traverse  every  cross-section  of  such  a  tube 
in  unit-time.  If  the  source  is  a  radiant  point  P  or  a  luminous  body 
of  such  relatively  minute  dimensions  that  it  may  be  considered  as 
physiologically  a  mere  point  (or  centre)  of  light,  the  light-tubes  will 
be  cones  with  their  vertices  at  the  point-source.  The  quantity  of 
light  radiated  in  a  given  time  from  a  steady  source  may  be  expressed 
generally  as  the  product  of  two  factors,  one  of  which  has  to  do  with 
the  purely  geometrical  relations,  whereas  the  other  depends  on  the 
physical  nature  and  condition  of  the  radiating  body.  Thus,  in  the 
simplest  case,  when  we  have  a  point-source  at  the  point  P,  the  quantity 
of  light  which  in  unit-time  "flows"  through  any  cross-section  of  an 
elementary  tube  of  light  may  be  represented  as  follows: 

dL  =  C  -  <fco,  (480) 

where  do)  denotes  the  magnitude  of  the  solid  angle  of  the  narrow  cone 
of  rays  emanating  from  P,  and  where  C  denotes  a  certain  magnitude 
called  the  "candle-power"  of  the  point-source  in  the  direction  of  the 
axis  of  the  cone.  If  around  P  as  centre  a  sphere  of  unit-radius  is 
described,  the  quantity  of  light  that  falls  on  a  unit  area  of  this  sphere 
will  be  numerically  equal  to  the  factor  here  denoted  by  C.  In  general, 
the  value  of  C  will  vary  with  the  directions  of  the  light-rays;  but  if 
we  may  assume  that  the  point-source  radiates  light-energy  at  approxi- 
mately the  same  rate  in  all  directions,  the  total  quantity  of  light- 
energy  per  second  that  traverses  any  closed  surface  surrounding  the 
point  P  will  be  equal  to  4irC. 

If  P'  designates  the  position  of  a  point  within  the  elementary  conical 
light-tube  of  solid  angle  du  which  lies  on  a  surface  a'  at  a  distance  from 
the  radiant  point  P  denoted  by  r  =  PP',  and  if  dvf  denotes  the  area 
of  the  surface-element  that  is  cut  out  of  the  surface  a'  by  the  cone, 
and,  finally,  if  <p'  denotes  the  acute  angle  between  the  normal  to  the 
surface  a'  at  the  point  P'  and  the  straight  line  PP',  then 

r2- d^  =  do-' -cos  <p'\ 

and  accordingly  we  can  "write: 

„  da'  -  cos  (p' 
dL  =  C-da  =  C .— -  ,  (481) 

where  dL  denotes  the  quantity  of  light  emanating  from  P  that  falls 
every  second  on  the  surface-element  dvf.  The  quantity  of  light- 
energy  which  is  received  by  unit-area  of  the  illuminated  surface  in 


§  388.]  Intensity  of  Illumination.  573 

unit-time  is  called  the  intensity  of  illumination  of  the  surface  a'  at 
the  point  P' \  and,  since  this  magnitude  is  defined  by  the  equation 

dL  cos  v 

M-C—T,  (482) 

we  see  that  the  intensity  of  illumination  is  inversely  proportional  to 
the  square  of  the  distance  from  the  point-source  and  directly  pro- 
portional to  the  cosine  of  the  angle  of  incidence  and  to  the  candle- 
power  of  the  source  in  the  given  direction. 

388.    Radiation  of  a  Luminous  Surface-Element. 

If  the  light-source  at  P  must  be  regarded  as  a  luminous  element  of 
surface  (da)  rather  than  as  a  mathematical  point,  the  quantity  of 
light-energy  dL  that  is  emitted  in  a  given  direction  in  the  unit  of  time 
will  depend  not  only  on  the  magnitude  of  da  but  also  on  the  angle  of 
emission  (<p)  between  the  normal  to  da-  at  P  and  the  given  direction 
PP' .  Thus,  according  to  LAMBERT'S  Law,  the  specific  energy  of  the 
radiation  of  the  luminous  surface-element  da  in  the  direction  PP' 
will  be  expressed  by  the  formula: 

C  =  i-da-cos<f>,  (483) 

where  the  co-efficient  i  denotes  a  magnitude  depending  on  the  physical 
nature  of  the  light-source  (for  example,  its  temperature,  radiating 
power,  etc.)  which  is  called  the  specific  intensity  or  the  intensity  of 
radiation  of  the  luminous  surface  a  at  the  point  P.  The  apparent 
uniformity  of  the  brightness  of  the  sun's  disc  is  in  agreement  with 
this  "cosine-law".  Thus,  near  the  margin  of  the  sun's  disc,  areas 
which  appear  to  be  of  the  same  size  as  areas  nearer  the  centre,  but 
which  in  reality  are  larger  than  their  oblique  projections,  do  not  radiate 
any  more  energy  than  the  smaller  but  more  central  areas  of  the  same 
apparent  size. 

Hence,  according  to  the  so-called  "cosine-law  of  emission",  the 
quantity  of  light-energy  radiated  per  unit  of  time  from  the  luminous 
surface-element  da  to  the  illuminated  element  da'  in  the  direction 
PP'is: 

.  da  •  da'  •  cos  <p  •  cos  <p' 
dL  =  i  -  —          — 2 —          ~~  •  (484) 

By  means  of  this  fundamental  formula  of  photometry,  due  originally 
to  LAMBERT/  the  factor  denoted  by  i  may  also  be  defined  as  the  quan- 

1  J.  H.  LAMBERT:  Photometric,  sive  de  mensura  et  gradibus  luminis  colorum  et  umbrae 
(Augsburg,  1760).  See  also  German  translation  by  E.  ANDING  in  Nos.  31-33  of  OSTWALD'S 
"  Klassiker  der  exakten  Wissenschaften  "  (Leipzig,  1892).  Also,  see  A.  BEER:  Grund- 
riss  des  photometrischen  Calcueles  (Braunschweig,  1854). 


574  Geometrical  Optics,  Chapter  XIV.  [  §  389. 

tity  of  light  which  in  the  unit  of  time  is  radiated  from  a  unit-area  of 
the  radiating  surface  to  another  unit-area  at'  unit-distance  from  it, 
when  the  line  PP'  is  a  common  normal  to  the  radiating  surface  at 
P  and  to  the  illuminated  surface  at  P'  .  As  a  matter  of  fact,  it  is 
found  by  experiment  that  the  specific  intensity  i  varies  with  the  angle 
of  emission  <p  and  according  to  a  peculiar  law  for  each  different  sub- 
stance; but  in  the  following  discussion  it  will  be  simpler  to  disregard 
this  variation  and  to  assume  therefore  that  the  value  of  i  is  independ- 
ent of  the  angle  <p. 

The  symmetry  of  the  expression  on  the  right-hand  side  of  the  above 
equation  cannot  fail  to  be  remarked.  Thus,  for  example,  the  quantity 
of  light  conveyed  from  da  to  da'  in  a  given  time  is  the  same  as  would 
be  transmitted  in  this  same  time  from  da'  to  da  in  case  the  roles  of 
the  two  surfaces  were  interchanged,  so  that  da'  was  the  radiating  ele- 
ment of  specific  intensity  equal  to  i  and  da  was  the  illuminated  element. 

Since 

da'  •  cos  <pr 


and  since,  also,  if  duf  denotes  the  solid  angle  subtended  at  P'  by  the 
radiating  surface-element  da, 

da  •  cos  < 


formula  (484)  may  be  written  likewise  in  either  of  the  two  following 
forms  : 

dL  =  i-da-  cos  <p  •  du  =  i-  da'  -  cos  <p'  -  du'.  (485) 

389.    Equivalent  Light-Source.    The  intensity  of  illumination  at  P' 
due  to  the  radiating  element  da  at  P,  viz., 

-7-7  =  i  -  cos  <p'  -  da',  (486) 

da 

is  proportional  to  the  specific  intensity  (i)  of  the  source  and  also  to 
the  solid  angle  (do/)  subtended  at  P'  by  the  radiating  surface-element 
(da)  and  the  cosine  of  the  angle  of  incidence  (<?')  of  the  rays.  With 
respect  to  the  illumination  at  P',  the  most  important  deduction  to  be 
made  here  is  that,  so  far  as  the  resultant  effect  at  P'  is  concerned,  the 
surface-element  da  may  be  supposed  to  be  replaced  by  its  central  pro- 
jection from  P'  on  to  any  other  surface  in  the  same  optical  medium, 
provided  we  ascribe  the  same  specific  intensity  i  to  the  corresponding 


§  390.]  Intensity  of  Illumination.  575 

points  of  the  projection-surface.1  Accordingly,  a  fictitious  source  of 
light,  or  rather  an  imaginary  distribution  of  the  specific  intensity,  can 
be  thus  substituted  in  place  of  the  actual  distribution  so  as  to  have 
precisely  the  same  effect  at  a  prescribed  point  P'.  However,  this 
so-called  equivalent  surface-distribution  of  the  specific  intensity  —  or 
"equivalent  light-source"  —will,  in  general,  produce  a  different  effect 
from  that  produced  by  the  actual  light-source  at  any  point  other  than 
the  given  point  P'  . 

ART.  122.     INTENSITY   OF  RADIATION  OF  OPTICAL  IMAGES. 

390.     Optical  System  of  Infinitely  Narrow  Aperture  (Paraxial  Rays). 

Let  M,  M'  designate  the  positions  of  a  pair  of  conjugate  axial  points 
of  a  centered  system  of  spherical  surfaces,  and  let  us  suppose,  at  first, 
that  the  aperture  of  the  system  is  infinitely  narrow,  so  that  only  the 
so-called  paraxial  rays  emanating  from  the  luminous  point-source  M 
can  traverse  the  system.  To  the  bundle  of  paraxial  rays  in  the  Object- 
Space  of  solid  angle  dco  corresponds  also  a  bundle  of  paraxial  rays  in 
the  Image-Space  of  solid  angle  du'  ;  and  if  C  denotes  the  candle-power 
of  the  point-source,  the  quantity  of  light  radiated  from  M  in  unit-time 
will  be  dL  =  C-dw;  and,  similarly,  the  quantity  of  light  radiated  in 
the  same  time  from  the  conjugate  image-point  Mf  will  be  dL'  =  C'  -duf, 
where  C'  denotes  the  candle-power  of  the  image-point  M'  regarded 
as  a  source  of  light  in  the  Image-Space.  Moreover,  for  the  sake  of 
simplicity,  let  us  assume  here  that  no  light-energy  is  "lost"  either  by 
absorption  in  traversing  the  various  media  or  by  undesirable  reflexions 
at  the  spherical  surfaces;  and  although  this  assumption  is  notoriously 
contrary  to  the  fact,  it  will  not  materially  affect  the  conclusions  which 
we  have  here  in  view.  Accordingly,  putting  dLf  =  dL,  we  obtain 
therefore  : 

C-du  =  C'-do'. 

The  following  relation  may  easily  be  deduced  : 


where  uk,  u'k  denote  the  abscissae  of  the  points  where  the  rays  cross 
the  optical  axis  before  and  after  refraction,  respectively,  at  the  &th 
surface  of  the  centered  system  of  m  spherical  surfaces.  If,  therefore, 

1  See  E.  ABBE:  Ueber  die  Bestimmung  der  Lichtstaerke  optischer  Instrumente:  Jen. 
Zft.f.  Med.  u.  Natw.,  vi  (1871),  263-291.  Also,  Gesammelte  Abhandlungen,  Bd.  I  (Jena, 
1904),  14-44- 


576  Geometrical  Optics,  Chapter  XIV.  [  §  391. 

F  denotes  the  lateral  magnification  of  the  system  with  respect  to  the 
pair  of  conjugate  axial  points  M,  Mf,  and  if  also  n,  n'  denote  the  indices 
of  refraction  of  the  media  of  the  Object-Space  and  Image-Space,  re- 
spectively, we  obtain  by  the  employment  of  formula  (93)  : 


u  n 

and,  hence: 


whereby,  knowing  the  candle-power  (C)  of  the  point-source  on  the 
axis  of  the  optical  system,  and  knowing  also  the  constants  of  the  sys- 
tem, we  are  enabled  to  determine  the  corresponding  candle-power  (  C') 
of  the  image-point  M'. 

If,  instead  of  a  point-source  at  the  axial  point  M,  we  have  a  luminous 
surface-element  dcr  at  right  angles  to  the  optical  axis  at  M,  the  image 
thereof  will  be  a  surface-element  da'  at  right  angles  to  the  optical  axis 
at  the  point  M',  of  such  dimensions  that 

d<r'  =    Y2-da. 
Hence,  since  here  we  have 

dL  =  i-da-du  =  dL'  =  i'-da'-da', 

where  *,  ir  denote  the  specific  intensities  in  the  direction  of  the  axis 
of  the  radiating  elements  da,  d<r',  respectively,  we  obtain  in  this  case 
the  following  striking  relation  : 

t-^ 
i  ~  n2' 

391.     Optical  System  of  Finite  Aperture. 

Finally,  let  us  now  proceed  to  the  more  general  case  and  assume 
that  the  aperture  of  the  optical  system  is  finite;  and  let  us  denote  by 
i  the  specific  intensity  of  radiation,  in  a  direction  defined  by  the  slope- 
angle  8,  of  a  luminous  surface-element  da-  placed  at  right  angles  to 
the  optical  axis  at  the  point  M.  The  quantity  of  light  radiated  in 
unit-time  from  the  element  da  to  an  elementary  annular  ring  of  the 
entrance-pupil  whose  inner  and  outer  radii  subtend  at  the  axial  object- 
point  M  angles  denoted  by  0  and  6  +  dO,  respectively,  may  be  easily 
calculated  from  the  fundamental  formula  (484)  and  will  be  found  to  be  ; 

dL  =  27ri-da'sm  8-d  (sin  6). 


§  392.]  Intensity  of  Illumination.  577 

Employing  the  same  symbols  with  primes  to  denote  the  corresponding 
magnitudes  in  the  Image-Space,  we  shall  find  also  a  precisely  analogous 
expression  for  the  quantity  of  light  that  is  radiated  per  unit  of  time 
from  the  image-element  dv'  to  the  corresponding  elementary  annular 
ring  of  the  exit-pupil,  viz.: 

dL'  =  2Trif'd<r'>sin  tf-d  (sin  tf). 

Now  if  da'  is  to  be  a  correct  image  of  the  object-element  da-,  it  is  neces- 
sary to  suppose  that  M,  M'  are  an  aplanatic  pair  of  points,  so  that 
the  Sine-Condition  is  satisfied,  whereby  we  must  have  (§  277): 

rt-sin  0  =  ri  '  •  F-sin  6'. 

Introducing  this  condition,  and  employing  here  also  the  relation 

da'  =    Y*'dv, 

and,  finally,  assuming,  as  before,  that  dL'  —  dL,  we  derive  again  the 
same  relation  as  above,  viz.  : 


Accordingly,  no  matter  how  the  specific  intensities  of  radiation  of 
object  and  image  may  vary  for  different  angles  of  emission,  their  ratio  is 
the  same  for  every  pair  of  values  of  6  and  0'.  This  constant  ratio  de- 
pends only  on  the  indices  of  refraction  of  the  media  in  which  the 
object  and  image  are  situated;  and  the  specific  intensity  of  radiation 
(if)  of  any  element  of  the  image  in  a  given  direction  (0')  is  equal  always  to 
(nf  /n)2  times  that  of  the  corresponding  object-element  in  the  conjugate 
direction  (0).1 

392.  In  deriving  the  above  resultte,  it  was  assumed  that  there  were 
no  losses  of  light  by  absorption,  reflexion,  etc.,  so  that  we  could  put 
dL'  —  dL.  It  would  have  been  more  correct  to  have  written: 

dL'  =  (i  -  i)dL, 
where  rj  denotes  the  fraction  of  the  original  quantity  of  light  that  is 

1  This  result  is  identical  with  KIRCHHOFF'S  well-known  law  of  radiation.  See  G. 
KIRCHHOFF:  Ueber  das  Verhaeltniss  zwischen  dem  Emissionsvermoegen  und  dem  Ab- 
sorptionsvermoegen  der  Koerper  fur  Waerme  und  Licht.  POGG.  Ann.,  cix  (1860),  275- 
301.  Also,  R.  CLAUSIUS:  Ueber  die  Concentration  vOn  Waerme-  und  Lichtstrahlen  und 
die  Grenzen  ihrer  Wirkung.  POGG.  Ann.,  cxxi  (1864);  also,  BROWNE'S  English  trans- 
lation of  CLAUSIUS'S  Mechanical  Theory  of  Heat  (London,  1879),  Chapter  XII. 

Starting  from  KIRCHHOFF'S  law  of  radiation,  HELMHOLTZ  deduced  the  Sine-Law;  see 
H.  HELMHOLTZ:   Die  theoretische  Grenze  fur  die  Leistungsfaehigkeit  der  Mikroskope: 
POGG.  Ann.  Jubelband,  1874,  557-584. 
38 


578  Geometrical  Optics,  Chapter  XIV.  [  §  393. 

dissipated  in  its  passage  through  the  system,  and  is  a  function  of  the 
angle  of  emission  (0),  which  may  be  determined  in  any  given  special 
case.  Under  these  circumstances,  formula  (488)  would  be  modified 
as  follows: 

*'  ,     x 

-T  =  (l  _  ^)  — 2  ;  (489) 

which  shows  also  that  the  ratio  i'/i  is  in  reality  a  function  of  the 
angle  0.1 

In  nearly  all  actual  optical  instruments  the  first  and  last  media  are 
both  air  (n  =  nf  =  i);  even  in  the  so-called  "immersion-systems" 
the  source  is  not  the  object  immersed  in  the  fluid,  inasmuch  as  the 
object  is  illuminated  from  without.  The  case  when  n'  >  n  is  hardly 
realizable.  Thus,  under  the  most  favourable  conditions,  the  specific 
intensity  of  radiation  from  a  definite  part  of  the  image  in  a  given  direct- 
ion will  always  be  less  than  the  specific  intensity  of  radiation  from  the 
corresponding  part  of  the  object  in  the  conjugate  direction.  For  ex- 
ample, the  intensity  of  radiation  of  the  sun's  image  at  the  focus  of  a 
convex  lens  can  never  be  greater  than  that  of  the  sun  itself,  although 
the  intensity  of  illumination  of  a  screen  placed  at  the  focus  of  the 
glass  may  be  much  greater  with  the  lens  than  without  it. 
393.  The  Illumination  in  the  Image-Space. 

The  image  M'Q'  of  a  luminous  object  M Q  may  be  regarded  as  the 
source  of  all  the  illumination  in  the  Image-Space ;  and  in  case  we  wish 

to  ascertain  the  intensity  of  the  illumina- 
tion   produced   at   any    point    Rr   of    the 
D  Image-Space  by  an  element  of  the  image 

at  P',  we  have  merely  to  trace  backwards 
Xl     through  the  optical  system  the  path  of  the 
image-ray  P'R'  and  thereby  determine  the 
point  P  of  the  object  that  is  conjugate  to 
the  image-point  P'.     The  specific  intensity 
of  the  radiation  from  P'  in  the  direction 
FIG- 17°-  P'R'  is  (n'/n)2  times  that  from  the  object- 

EXIT-PUPIL   AS    EQUIVALENT     pojnt  p  jn  tjie  conjugate  direction  in  the 

LUMINOUS  SURFACE.  *1     .  J    c 

Object-Space;    provided    we   assume   that 
none  of  the  light  is  dissipated  in  its  passage  through  the  system. 

The  part  of  the  image  M'Q'  (Fig.  170)  that  is  effective  in  producing 
illumination  at  a  point  R'  of  the  Image-Space  is  easily  found  by  pro- 
jecting the  exit-pupil  C'D'  on  to  the  image-plane  cr';  thus,  in  the 

1  See  S.  CZAPSKI:  Theorie  der  optischen  Instrumente  nach  ABBE  (Breslau,  1893),  p.  179. 


§  394.]  Brightness.  579 

diagram,  U'V  represents  the  effective  part  of  the  image  with  respect  to 
the  illumination  at  the  point  R'  '.  In  place  of  the  portion  of  the  image 
U'V',  we  may  substitute  an  equivalent  distribution  of  light  (§  389)  by 
considering  the  specific  intensity  of  the  parts  of  the  image  comprised 
between  U'  and  V  as  localized  at  the  corresponding  parts  of  the  exit- 
pupil  ;  and  this  distribution  of  light  supposed  to  be  spread  over  the 
exit-pupil  would  produce  exactly  the  same  effect  at  R'  as  is  produced 
there  by  the  image  of  the  luminous  object.  This  ingenious  method, 
due  to  ABBE/  enables  us  to  determine  the  intensity  of  illumination 
at  any  point  of  the  image  itself.  For  example,  the  nearer  the  point 
R'  is  to  the  point  Pf  of  the  image,  the  smaller  will  be  the  circular  space 
around  P'  that  is  obtained  by  projecting  the  exit-pupil  on  to  the  plane 
of  the  image;  and,  finally,  when  the  point  R'  coincides  with  P',  so 
that  the  exit-pupil  is  projected  on  to  the  image-plane  in  the  point  P' 
itself,  the  intensity  of  the  illumination  at  Pf  can  be  found  by  regarding 
the  illumination  there  as  due  to  a  distribution  of  light  over  the  exit- 
pupil  of  the  same  specific  intensity  of  radiation  as  that  of  the  point 
P',  viz.,  (n'/n)2i,  where  i  denotes  the  specific  intensity  of  radiation  in 
any  given  direction  (0)  of  the  object-point  P  conjugate  to  P'. 

ART.  123.     BRIGHTNESS   OF   OPTICAL  IMAGES. 

394.    Brightness  of  a  Luminous  Object. 

In  connection  with  the  definition  of  the  objective  intensity  of  illumi- 
nation (§  389)  at  a  given  place  of  an  illuminated  surface,  we  can  derive 
also  an  idea  of  what  is  meant  by  the  Brightness  of  the  source  as  seen 
by  an  eye  situated  at  the  place  in  question.  The  brightness  of  an 
element  da  of  a  radiating  surface  is  defined  as  the  quantity  of  light- 
energy  which  in  the  unit  of  time  falls  on  unit-area  of  the  image  d<r' 
that  is  formed  on  the  retina  of  the  eye  ;  in  other  words,  it  is  the  inten- 
sity of  illumination  of  the  element  of  the  retina-surface  that  is  affected 
by  the  given  element  of  the  luminous  body.  Thus,  if  dL  denotes  the 
quantity  of  light  which  is  radiated  per  unit  of  time  from  the  element 
da  into  the  eye,  the  brightness  of  this  element  is  defined  by  the  equa- 
tion: 

dL 


If  we  assume  that  there  is  no  loss  of  light  in  traversing  the  optical 

1  E.  ABBE:  Ueber  die  Bestimmung  der  Lichtstaerke  optischer  Instrumente.  Jen.  Z.ft. 
}.  Med.  u.  Nalw.,  vi  (£871),  263-291.  Also  Gesammelte  Abhandlungen,  Bd.  I  (Jena, 
1904),  14-44- 


580  Geometrical  Optics,  Chapter  XIV.  [  §  395. 

media  of  the  eye,  then 

2iri'  •  dff'  •  sin  0'  •  d  (sin  0') 

will  be  the  quantity  of  light  that  is  radiated  per  unit  of  time  across  an 
elementary  annular  ring  of  the  exit-pupil  of  the  eye,  where  0'  denotes 
the  angle  subtended  at  the  retina  by  the  inner  radius  of  this  ring  and 
i'  denotes  the  specific  intensity  of  the  image  da'  on  the  retina;  and 
hence  the  total  quantity  of  light  that  enters  the  pupil  will  be 

dL  =  iri'  -da  -sin2 9^, 

where  9^  denotes  the  angle  subtended  at  the  image  on  the  retina  by 
the  radius  of  the  exit-pupil  of  the  eye,  which  usually  does  not  exceed 
about  5°.  If,  therefore,  the  object  is  viewed  by  the  unaided  eye,  we 
find  for  the  so-called  natural  brightness  (B0)  of  the  luminous  surface- 
element  da  (supposed  to  be  situated  in  air,  so  that  n  =  i) : 

BO  =  fa,  =  7r-«'2-i-sm29o,  (491) 

where  i  denotes  the  specific  intensity  of  radiation  of  the  source,  and 
nr  denotes  the  refractive  index  of  the  vitreous  humour  of  the  eye.  It 
follows  immediately  from  this  expression  that  the  natural  brightness 
of  a  uniformly  radiating  surface  depends  only  on  the  intensity  of 
radiation  of  the  light-source  and  is  entirely  independent  of  the  dis- 
tance of  the  luminous  object  from  the  eye ;  as  is  found  to  be  practically 
the  case. 

i,  395.  In  the  next  place,  let  us  suppose  that  this  same  object  is 
viewed  through  an  optical  instrument  by  an  eye  placed  at  the  exit- 
pupil  of  the  instrument.  Everything  is  the  same  as  before,  except 
that  now,  instead  of  the  mere  optical  system  of  the  eye,  we  have  a 
compound  optical  system  formed  by  the  combination  of  the  eye  with 
the  optical  instrument.  If  we  disregard  all  losses  of  light  by  reflexion 
and  absorption,  and  assume,  as  before,  that  the  luminous  object  is  in 
air,  the  brightness  B  of  the  optical  image  as  seen  through  the  instru- 
ment will  be  equal  to  the  natural  brightness  BQ  of  the  object  as  viewed 
by  the  naked  eye,  provided  the  exit-pupil  of  the  eye  is  smaller  than 
that  of  the  instrument.  But  if,  on  the  other  hand,  the  diameter  of 
the  exit-pupil  of  the  instrument  is  smaller  than  that  of  the  eye,  the 
aperture-angle  will  be  an  angle  9'  <  9^,  so  that  in  this  case  we  shall 
have: 

B:BQ  =  sin2  9':  sin2  Q'O. 

Since  the  angles  9^,,  9'  are  so  small  that  we  may  substitute  the  tangents 


§  396.]  Brightness.  581 

of  these  angles  in  place  of  their  sines,  and  since,  moreover,  the  exit- 
pupil  of  the  eye  coincides  very  nearly  with  the  eye-pupil  (or  iris), 
we  obtain: 

B:B0  =  p'*:pl  (492) 

\ 

where  p0,  p'  denote  the  radii  of  the  iris-opening  and  exit-pupil 
of  the  instrument,  respectively;  so  that  the  brightness  of  the  image 
compared  with  the  natural  brightness  of  the  object  is  diminished  in 
the  ratio  of  the  size  of  the  exit-pupil  of  the  instrument  to  the  size  of 
the  eye-pupil.  It  is,  therefore,  impossible  by  means  of  any  optical 
instrument  to  increase  the  natural  brightness  of  an  object  as  seen  by 
the  unaided  eye.  Thus,  the  only  function  of  an  optical  instrument  is 
by  means  of  a  light-source  either  of  small  dimensions  or  very  far  away 
to  produce  an  effect  equal  to  that  which  could  be  produced  without 
the  instrument  only  by  a  larger  or  nearer  source  of  light  radiating  with 
equal  specific  intensity.1 

396.     Brightness  of  a  Point-Source. 

If  the  luminous  object  is  so  small  or  so  far  away  that  it  has  no 
sensible  apparent  size,  the  definition  of  brightness  given  above  (§  394) 
ceases  to  have  any  meaning;  for  the  image  on  the  retina  of  the  eye 
will  in  this  case  be  itself  a  mere  point  without  appreciable  area.  If, 
therefore,  the  source  of  illumination  is  a  point,  for  example,  a  fixed 
star,  the  brightness  is  defined  as  equal  or  proportional  to  the  quantity 
of  light  which  comes  to  us  from  it.  Thus,  when  we  speak  of  a  star 
of  the  "first  magnitude",  this  expression  refers  merely  to  the  amount 
of  light  we  receive  from  it  and  has  nothing  to  do  with  the  size  of 
the  star. 

If  in  formula  (481)  we  put  da'  =  irp\  (where  p0  denotes  the  radius 
of  the  eye-pupil)  and  cos  <p'  =  i  (since  the  rays  are  supposed  to  fall 
normally  on  the  retina  when  the  eye  is  directed  towards  the  point- 
source),  we  obtain: 

B-Orff.      _  (493) 

Hence,  the  brightness  of  an  object  which  appears  like  a  point  is  inversely 
proportional  to  the  square  of  its  distance  from  the  eye,  and  directly  pro- 
portional to  the  size  of  the  eye-pupil. 

Thus,  stars  which  are  invisible  to  the  naked  eye  may  be  brought  to 

1  Lord  RAYLEIGH,  in  his  brilliant  article  on  Optics  in  the  ninth  edition  of  the  Encyclo- 
paedia Britannica,  has  pointed  out  that  "  the  general  law  that  the  apparent  brightness 
depends  only  on  the  area  of  the  pupil  filled  with  light  "  was  stated  and  demonstrated  by 
ROBERT  SMITH.  See  SMITH'S  Optics  (Cambridge,  1738),  Vol.  I,  Sections  255  and  261. 


582  Geometrical  Optics,  Chapter  XIV.  [  §  396. 

view  by  the  aid  of  a  telescope,  whereby  the  eye  receives  a  greater 
quantity  of  light  from  the  star  than  before,  so  that  the  brightness 
(in  this  latter  sense  of  the  term)  is  increased;  whereas,  on  the  other 
hand,  the  brightness  of  the  background  of  the  sky  (using  the  word 
"brightness"  in  its  /original  sense,  as  defined  in  §  394)  will  be  di- 
minished. This  is  the  reason  why  a  powerful  telescope,  of  large 
aperture  and  great  magnifying  power,  may  enable  an  observer  to 
view  the  stars  even  in  the  noon-day  glare. 


APPENDIX. 

EXPLANATIONS   OF  LETTERS,   SYMBOLS,   ETC. 

The  meanings  of  the  principal  letters  and  symbols  both  in  the  text 
and  in  the  diagrams  are  here  set  forth  as  briefly  as  possible;  but  such 
uses  as  are  occasional  or  merely  incidental  are  generally  not  noted  at 
all.  In  consulting  these  tables,  it  is  important  to  bear  in  mind  this 
last  statement. 

I.     DESIGNATIONS   OF   POINTS  IN   THE   DIAGRAMS. 

As  a  rule  (but  not  without  exception),  the  positions  of  the  points 
in  the  diagrams  are  designated  by  Latin  capital  letters.  The  most 
important  uses  of  these  letters  are  explained  below. 

A 

1.  A,  A'  are  used  to  designate  the  primary  and  secondary  principal 
points,  respectively,  of  two  collinear  space-systems;  see  Fig.  92.     Simi- 
larly, as  in  Fig.  99,  Ak1  A'k  designate  the  principal  points  of  the  &th 
component  of  a  compound  optical  system. 

In  Chap.  XIII,  A,  A'  and  A,  A'  designate  the  positions  on  the 
optical  axis  of  the  two  pairs  of  principal  points  of  the  system  for  rays 
of  light  of  wave-lengths  X  and  X,  respectively. 

In  the  case  of  two  centrally  collinear  plane- fields ',  A  designates  the 
position  of  the  point  of  intersection  with  the  axis  of  collineation  (y) 
of  the  self-corresponding  ray  (#,  x')  that  meets  this  axis  at  right  angles. 

2.  Especially,  the  letter  A  is  used  to  designate  the  vertex  of  a  spher- 
ical refracting  (or  reflecting)  surface.     Similarly,  A  may  be  used  to 
designate  the  position  of  the  foot  of  the  perpendicular  let  fall  on  to  a 
plane  refracting  (or  reflecting)  surface  from  a  point  on  the  incident 
ray  regarded  as  object-point,  as  in  Fig.  8. 

The  letter  A  designates  the  optical  centre  of  an  Infinitely  Thin  Lens. 

The  vertex  of  the  kth  surface  of  a  centered  system  of  spherical 
surfaces  is  designated  by  Ak\  also,  the  optical  centre  of  the  kth  lens 
of  a  centered  system  of  Infinitely  Thin  Lenses. 

3.  In  Chap.  I X,  in  the  determination  of  the  path  of  a  ray  refracted 
obliquely  at  a  spherical  surface,  Ag,  A{  are   used  to   designate  the 
points  of  intersection  with  the  surface  of  the  radii  drawn  through  the 
points  designated  by  G  (14)  and  /  (18),  respectively  (Fig.  122).     In 

583 


584  Geometrical  Optics,  Appendix. 

Chap.  X,  in  the  case  of  a  ray  refracted  obliquely  through  a  centered 
system  of  spherical  surfaces,  A9t1ty  Aitk  have  the  same  meanings  as 
above  with  respect  to  the  kth  surface. 

B,B 

4.  B,  B'  are  used  to  designate  the  points  of  intersection  of  a  pair 
of  conjugate  rays  with  the  principal  planes  of  two  collinear  space- 
systems.     In  Fig.  99,  for  example,  Bk,  B'k  designate  the  points  where 
a  meridian  ray  crosses  the  principal  planes  of  the  kth  component  of 
a  compound  optical  system. 

In  particular,  B  designates  the  point  of  intersection  of  a  ray  with 
the  axis  of  collineation  (y)  of  two  centrally  collinear  plane-fields. 

5.  Especially,  B  designates  the  position  on  the  refracting  (or  re- 
flecting) surface  of  the  incidence-point  of  a  ray.     In  the  case  of  a 
centered  system  of  spherical  surfaces  or  a  prism-system,  Bk  designates 
the  incidence-point  of  the  ray  at  the  kth  surface. 

In  Chap.  XIII,  BkJ  Bk  designate  the  incidence-points  at  the  kth 
spherical  surface  of  rays  of  light  of  wave-lengths  X,  X,  respectively, 
whose  paths  in  the  Object-Space  are  identical. 

If  we  are  concerned  with  a  pair  of  rays  from  two  different  sources, 
whose  paths  lie  in  the  plane  of  a  principal  section,  their  incidence- 
points  may  be  designated  by  B  and  B  (or  by  Bk  and  Bk),  as,  for 
example,  in  Chap.  VIII. 

Usually,  however,  B  or  Bk  designates  the  position  of  the  incidence- 
point  of  the  chief  ray  of  a  bundle. 

C 

6.  C  is  used  primarily  to  designate  the  centre  of  a  spherical  refracting 
or  reflecting  surface.     The  centre  of  the  kth  surface  of  a  centered 
system  of  spherical  surfaces  is  designated  by  Ck. 

This  letter  is  used  also  to  designate  the  centre  of  collineation  of  two 
centrally  collinear  plane-fields,  as  in  Fig.  66. 

7.  In  Chap.  XIV,  C,   C'  are  used  to  designate  corresponding  ex- 
tremities of  conjugate  diameters  of  the  entrance-pupil  and  exit-pupil, 
respectively,  of  an  optical  system.      C  designates  the  lower  extremity 
of  the  diameter,  in  the  meridian  plane,  of  the  entrance-pupil. 

D,D 

8.  D  designates  the  foot  of  the  perpendicular  BD  let  fall  from  the 
incidence-point  B  on  to  the  optical  axis;  in  a  centered  system  of 
spherical  surfaces,  Dk  designates  the  foot  of  the  perpendicular  let  fall 
on  to  the  optical  axis  from  the  point  Bk  where  the  ray  meets  the  kth 
surface. 


Designations  of  Points.  585 

The  foot  of  the  perpendicular  let  fall  on  to  the  optical  axis  from  the 
point  B  (see  5)  is  designated  by  D,  as  in  Fig.  140. 

The  feet  of  the  perpendiculars  let  fall  on  to  the  optical  axis  from  the 
points  Bk,  Bk  (see  5)  are  designated  by  Dk,  Dk,  respectively. 

9.  D,  Df  are  used  (in  Chap.  XIV)  to  designate  corresponding  ex- 
tremities of  conjugate  diameters  of  the  entrance-pupil  and  exit-pupil, 
respectively,  of  an  optical  system.     Generally,  D  designates  the  upper 
extremity  of  the  diameter,  in  the  meridian  plane,  of  the  entrance- 
pupil  (see  7). 

10.  In  certain  of  the  prism-diagrams  of  Chap.  IV,  D  designates 
the  point  of  intersection  of  the  incident  and  emergent  rays. 

E 

11.  E,  E'  are  used  to  designate  the  infinitely  distant  point  of  the 
optical  axis  (x)  in  the  Object-Space  and  its  conjugate  point  in  the 
Image-Space,  respectively.    E'  designates,  therefore,  the  secondary  focal 
point  of  the  optical  system. 

E',  Ef  designate  the  secondary  focal  points  of  the  optical  system 
for  rays  of  light  of  wave-lengths  X,  X,  respectively. 

E'k  is  used  to  designate  the  secondary  focal  point  of  the  kth  com- 
ponent of  a  compound  optical  system,  as  in  Fig.  99. 

F 

12.  F,  F'  designate  the  primary  focal  point  in  the  Object-Space 
and  the  infinitely  distant  point  of  the  optical  axis  (#')  in  the  Image- 
Space,  respectively. 

F,  F  designate  the  primary  focal  points  of  the  optical  system  for 
rays  of  light  of  wave-lengths  X,  X,  respectively. 

Fh  designates  the  primary  focal  point  of  the  kih  component  of  a 
compound  optical  system,  as  in  Fig.  99. 

G 

13.  G  designates  the  incidence- point  of  a  ray  in  the  meridian  section 
of  an  infinitely  narrow  bundle  of  rays  (Fig.  127). 

14.  In  Chap.  IX,  G,  G'  are  used  to  designate  the  points  where 
an -oblique  ray  crosses  the  plane  of  the  principal  section  (rry-plane)  of 
the  spherical  refracting  surface,  before  and  after  refraction,  respect- 
ively (Figs.  122  and  123). 

H 

15.  H,  Hf  designate  the  points  where  an  oblique  ray  crosses  the 
central  transversal  plane  (yz-plane),  before  and  after  refraction,  respect- 


586  Geometrical  Optics,  Appendix. 

ively,  at  a  spherical  surface  (Fig.  123).  Similarly,  Hk,  H'k  are  used 
as  above  stated,  with  respect  to  the  kth  surface  of  a  centered  system 
of  spherical  surfaces. 

In  particular,  H,  Hf  designate  the  points  where  a  ray,  lying  in  the 
principal  section  of  a  spherical  refracting  surface,  crosses  the  central 
perpendicular,  before  and  after  refraction,  respectively  (Fig.  120). 


16.  In  certain  diagrams  in  Chapters  V,  VI  and  VII,  J,  /'  are  used 
to  designate  the  infinitely  distant  point  of  an  object-ray  s  and  the 
"Flucht"  Point  of  the  conjugate  image-ray  s',  respectively,  of  two 
collinear  plane-fields.     See,  for  example,  Fig.  67. 

17.  Especially,  in  Chap.  XI,  in  connection  with  the  theory  of  the 
refraction  of  an  infinitely  narrow  bundle  of  rays  at  a  spherical  surface, 
/,  I'  designate  the  infinitely  distant  point  of  the  range  of  primary 
object-points  lying  on  the  chief  incident  ray  and  the  "Fluent"  Point 
of  the  conjugate  range  of  primary  image-points  lying  on  the  chief 
refracted  ray,  respectively.     Thus,  /'  designates  the  secondary  focal 
point  of  the  collinear  plane-fields  of  the  meridian  sections  of  the  bundles 
of  incident  and  refracted  rays.     See  Fig.  128. 

Similarly,  I,  T  designate  the  infinitely  distant  point  of  the  range 
of  secondary  object-points  lying  on  the  chief  incident  ray  and  the 
"Flucht"  Point  of  the  conjugate  range  of  secondary  image-points  lying 
on  the  chief  refracted  ray,  respectively.  Thus,  also  /'  designates  the 
secondary  focal  point  of  the  collinear  plane-fields  of  the  sagittal  sections 
of  the  bundle  of  incident  and  refracted  rays.  See  Fig.  128. 

According,  therefore,  as  the  chief  incident  ray  is  regarded  as  the 
base  of  a  range  of  primary  or  secondary  object-points,  the  infinitely 
distant  point  of  this  ray  is  designated  by  /  or  I. 

Moreover,  Ik,  Tk  designate  the  focal  points  of  the  systems  of  meridian 
and  sagittal  rays,  respectively,  with  respect  to  a  given  chief  ray  re- 
fracted at  the  kth  surface  of  a  centered  system  of  spherical  surfaces. 

18.  In  Chap.  IX,  7,  I'  designate  the  points  where  an  oblique  ray 
crosses  the  (horizontal)  meridian  xz-plane,  before  and  after  refraction, 
respectively,  at  a  spherical  surface  (Fig.  122). 

Similarly,  in  Chap.  X,  Ik,  Ik  (or  Ik+l)  are  used  in  the  same  way 
as  above,  with  respect  to  the  kth  surface  of  a  centered  system  of 
spherical  refracting  surfaces. 

J 

19.  In  certain  diagrams  in  Chapters  V,  VI  and  VII,  /,  /'  designate 
the  "Flucht"  Point  of  an  object-ray  5  and  the  infinitely  distant  point 


Designations  of  Points.  587 

of  the  conjugate  image-ray  s',  respectively,  of  two  collinear  plane- 
fields.     See,  for  example,  Fig.  67. 

20.  In  Chap.  XI,  in  connection  with  the  theory  of  the  refraction 
of  an  infinitely  narrow  bundle  of  rays  at  a  spherical  surface,  /,  /' 
designate  the  "FluchC  Point  of  the  range  of  primary  object-points 
lying  on  the  chief  incident  ray  and  the  infinitely  distant  point  of  the 
corresponding  range  of  primary  image-points  lying  on  the  chief  re- 
fracted ray,  respectively.     Thus,  /  designates  the  primary  focal  point 
of  the  collinear  plane-fields  of  the  meridian  sections  of  the  bundles  of 
incident  and  refracted  rays  (Fig.  I28J. 

Similarly,  J,  J'  designate  the  "Fluent"  Point  of  the  range  of  sec- 
ondary object-points  lying  on  the  chief  incident  ray  and  the  infinitely 
distant  point  of  the  corresponding  range  of  secondary  image-points 
lying  on  the  chief  refracted  ray,  respectively.  Thus,  also,  J  designates 
the  primary  focal  point  of  the  collinear  plane-fields  of  the  sagittal 
sections  of  the  bundles  of  incident  and  refracted  rays  (Fig.  128). 

According,  therefore,  as  the  chief  refracted  ray  is  regarded  as  the 
base  of  a  range  of  primary  or  secondary  image-points,  the  infinitely 
distant  point  of  this  ray  is  designated  by  J'  or  /'. 

Moreover,  Jk,  Jk  designate  the  primary  focal  points  of  the  systems 
of  meridian  and  sagittal  rays,  respectively,  with  respect  to  a  given 
chief  ray  refracted  at  the  &th  surface  of  a  centered  system  of  spherical 
surfaces. 

K 

21.  In  Fig.  128,  K  designates  the  centre  of  perspective  of  the  range 
of  object-points  lying  on  the  chief  incident  ray  and  the  range  of  pri- 
mary image-points  lying  on  the  corresponding  refracted  ray. 

22.  In  Chap.   XII,    K,   K  and   K',   K'  designate  the  centres  of 
curvature  of  the   two  astigmatic  image-surfaces,  before  and  after  re- 
fraction, respectively,  at  one  of  a  centered  system  of  spherical  surfaces 
(Figs.  141,  142). 

L,L 

23.  L,  L'  designate  the  points  where  a  ray,  lying  in  the  principal 
section  of  a  spherical  refracting  surface,  crosses  the  axis,  before  and 
after  refraction,  respectively  (Fig.  120). 

L,  L'  designate  the  points  where  the  chief  ray  of  a  bundle  crosses 
the  axis,  before  and  after  refraction,  respectively,  at  a  spherical  surface. 

In  certain  cases,  also,  L,  L'  (or  L,  L')  are  used  to  designate  the 
points  where  an  object-ray  and  the  corresponding  image-ray,  respect- 
ively, cross  the  optical  axis  of  a  centered  system  of  spherical  refracting 
surfaces. 


588  Geometrical  Optics,  Appendix. 

Sometimes  L"  is  used  to  designate  the  point  where  a  second  ray 
emanating  from  the  axial  object-point  L  crosses  the  axis  after  emerging 
from  the  optical  system  (Fig.  117). 

L'k  (or  Lk+l)  designates  the  point  where  a  ray  lying  in  the  prin- 
cipal section  crosses  the  optical  axis  after  refraction  at  the  &th  surface 
of  a  centered  system  of  spherical  surfaces.  If  the  ray  is  the  chief  ray 
of  the  bundle,  the  point  in  question  is  designated  by  L'k  (or  Lk+l)> 
The  point  where  the  ray  crosses  the  axis  in  the  Object-Space  is  desig- 
nated by  L!  (or  LJ. 

M,  M,  W 

24.  M,  M'  designate  a  pair  of  conjugate  axial  points  of  an  optical 
system;  especially,  a  pair  of  points  where  a  paraxial  ray  crosses  the 
optical  axis  in  the  Object-Space  and  Image-Space,  respectively. 

In  Chap.  XIII,  M',  M'  and  3ft'  designate  the  points  where  paraxial 
rays  of  light  of  wave-lengths  X,  X  and  I,  respectively,  all  emanating 
originally  from  the  same  axial  object-point  M,  cross  the  optical  axis 
in  the  Image-Space. 

M'k  (or  Mk+l)  designates  the  point  where  a  paraxial  ray,  emanat- 
ing from  the  axial  object-point  Mlt  crosses  the  optical  axis  of  a  centered 
system  of  spherical  surfaces  after  refraction  at  the  &th  surface  (see 
Fig.  71).  Here  also  M'k  has  a  meaning  corresponding  to  that  of  M' 
above. 

25.  Especially,  M,  M'  designate  the  points  where  the  optical  axis 
crosses  the  transversal  object-plane  (cr)  and  the  conjugate  (GAUSsian) 
image-plane  (O ;  or  the  points  where  the  optical  axis  crosses  the  focus- 
plane  and  the  screen-plane,  respectively  (Chap.   XIV).     See  69.     M 
may  be  defined  as  the  foot  of  the  perpendicular  let  fall  on  to  the 
optical  axis  from  the  extra-axial  object-point  Q-,  and  if  Q'  designates 
the  GAUSsian  image-point  corresponding  to  Q,  Mr  will  designate  also 
the  foot  of  the  perpendicular  let  fall  on  to  the  optical  axis  from  Q1 '. 

26.  M,  M'  designate  a  second  pair  of  conjugate  axial  points,  with 
respect  either  to  a  single  spherical  surface  or  a  centered  system  of 
spherical  surfaces.     The  meanings  of  M1 ',  M' ',   M'k  (or  Af&+1);   and 
M'k  correspond  exactly  with  the  meanings  given  above  (24)  of  M ',  M' ; 
M'k  (or  Mk+}) ;  and  M'k,  respectively. 

27.  Especially,  M,  M'  designate  the  positions  on  the  axis  of  the 
centres  of  entrance-pupil  and  exit-pupil,  respectively,  of  the  optical 
system.     In  an  optical  system  of  m  centered  spherical  surfaces,  the 
centres  of  the  pupils  may  be  designated  by  M {  and  M '„. 

If  M,  M'  designate  the  pupil-centres  of  the  optical  system  for  rays 
of  wave-length  X,  M,  M'  may  be  used  to  designate  the  pupil-centres 
for  rays  of  wave-length  X. 


Designations  of  Points.  589 

28.  In  an  Infinitely  Thin  Lens,  M,  M'  are  used  (as  in  Fig.  75)  to 
designate  the  points  where  a  paraxial  ray  crosses  the  optical  axis, 
before  and  after  passing  through  the  lens.     And,  especially,  in  the 
case  of  a  centered  System  of  Infinitely  Thin  Lenses,  Mk,  M'k  are  used 
in  this  way  with  respect  to  the  &th  lens.     Exactly,  the  same  statements 
can  be  made  here  with  reference  to  the  use  of  M ,  M'  and  Mk,  M'k. 

29.  M"  is  used  in  various  ways;  for  example,  to  designate  the 
point  where  the  focussing  screen  is  crossed  by  the  optical  axis,  or,  as 
in  Fig.  161  (and  elsewhere),  to  designate  the  centre  of  the  "blur-circle" 
corresponding  to  an  axial  object-point  M. 

N 

30.  N,  Nf  are  used  to  designate  points  on  the  normal  to  a  refracting 
surface  at  the  incidence-point   B  in  the  first  and  second  medium, 
respectively  (as  in  Fig.  5). 

31.  Especially,    N,    N'  designate  the  pair  of  nodal  points  of  an 
optical  system  (Fig.  92). 

O 

32.  0  designates  the  position  on  the  optical  axis  of  the  centre  of 
the  aperture-stop. 

33.  In  the  prism-diagrams  of  Chap.  IV,  0  designates  the  point  of 
intersection  of  the  normals  to  the  two  faces  of  the  prism  at  the  points 
of  entry  and  exit.     In  the  case  of  a  train  of  prisms,  a  numerical  sub- 
script indicates  the  prism  to  which  the  letter  refers  (as  in  Fig.  45). 

34.  0  is  used  also  to  designate  the  position  of  the  optical  centre  of 
a  thick  lens  (Fig.  74). 

35.  In  Chapters  V,  VI  and  VII,  0,  0'  occur  frequently  to  designate 
a  pair  of  conjugate  points. 

p,p 

36.  P,  Pf  designate  the  point  where  the  object-ray  crosses  the 
transversal  object-plane  a  and  the  point  where  the  corresponding  re- 
fracted ray  crosses  the  transversal  image-plane  a',  respectively  (69). 

P'k  (or  Pk+l)  designates  the  point  where  a  ray  crosses  the  trans- 
versal plane  <j'k  after  refraction  at  the  &th  surface  of  a  centered  system 
of  spherical  surfaces. 

Pl  designates  the  point  where  the  rectilinear  path  of  the  ray  in  the 
Object-Space  crosses  the  transversal  object-plane  ffl ;  especially,  it  des- 
ignates the  position  of  the  object-point  in  this  plane,  and,  in  general, 
the  same  extra-axial  object-point  as  is  designated  by  Ql  (see  39). 
In  a  certain  sense  (see  Chap.  XII)  the  point  P'm  may  be  regarded  as 
the  image  of  the  object-point  PP 


590  Geometrical  Optics,  Appendix. 

37.  P,  P'  are  used  in  Chap.  XII  to  designate  the  point  where  the 
object-ray  crosses  the  transversal  plane  a  in  the  Object-Space  and  the 
point  where  the  corresponding  image-ray  crosses  the  conjugate  plane 
<r'  in  the  Image-Space,  respectively  (71).     The  object-ray  here  men- 
tioned is  a  ray  that  goes  through  the  point  P  (36). 

Similarly,  also,  P'k  (or  Pk+l)  designates  the  point  where  a  ray 
which  in  the  Object-Space  goes  through  P1  crosses  the  transversal 
plane  o^  after  refraction  at  the  kth  surface  of  a  centered  system  of 
spherical  surfaces.  The  point  where  the  object-ray  crosses  the  first 
one  (o^)  of  this  series  of  transversal  planes  is  designated  by  P1  (or  Q,). 

38.  P,  P';  Q,  Q';  R,  #';  S,  S'  and  P,  P';  Q,  p';  R,  R';  3,  3' 
are  used  frequently  to  designate  pairs  of  corresponding  points  of  pro- 
jective  point-ranges.     Thus,  for  example,  in  Chap.  XI,  P,  P'  designate 
a  pair  of  corresponding  points  of  the  ranges  of  primary  object-points 
and  image-points  lying  along  the  chief  incident  ray  of  a  narrow  bundle 
of  rays  and  the  corresponding  refracted  ray,  respectively;  and,  sim- 
ilarly, F,  P'  designate  corresponding  points  of  the  ranges  of  secondary 
object-points  and  image-points  lying  along  the  same  chief  incident 
and  refracted  rays,  respectively. 

Q,Q 

39.  Q,  Q'  designate  a  pair  of  conjugate  points,  especially  a  pair 
of  extra-axial  conjugate  points,  of  two  collinear  systems. 

In  general,  Q,  Q'  designate  a  pair  of  points,  lying  outside  the  axis 
of  the  optical  system,  which,  by  GAUSS'S  Theory,  are  conjugate  to 
each  other,  with  respect  to  either  a  single  spherical  surface  or  a  centered 
system  of  spherical  surfaces.  Especially,  Q,  Q'  designate  a  pair  of 
conjugate  points  lying  in  the  transversal  planes  o-,  a',  respectively. 

<2,  ()'  are  used  also  (Chap.  XIV)  to  designate  the  centres  of  the 
"blur-circles"  in  the  focus-plane  (cr)  and  the  screen-plane  (a') ,  respect- 
ively. 

In  case  we  have  to  do  with  rays  of  light  of  two  different  colours 
(as  in  Chap.  XIII),  Q' ',  "Qr  designate  the  points  conjugate  to  Q  for 
rays  of  light  of  wave-lengths  X,  X,  respectively. 

Qk  (°r  Qk+\)  designates  the  point  where,  according  to  GAUSS'S 
Theory,  a  ray,  emanating  originally  from  the  object-point  Qv  crosses 
the  transversal  plane  ak  after  refraction  at  the  kth  surface  of  a  centered 
system  of  spherical  surfaces. 

40.  The  point  where  an  object-ray,  which  goes  through  the  object- 
point  Q  (or  P),  crosses  the  plane  (<r)  of  the  entrance-pupil  is  desig- 
nated by  Q  or  P;  and  the  point  in  the  plane  (<r')  of  the  exit-pupil, 
which,  by  GAUSS'S  theory,  is  conjugate  to  Q  (or  P),  is  designated  by  Q'. 


Designations  of  Points.  591 

In  a  centered  system  of  spherical  surfaces,  Qt  (or  P^  designates  the 
point  where  an  object-ray,  containing  the  object-point  Ql  (or  PJ, 
crosses  the  plane  c^  of  the  entrance-pupil;  and  Q^  (or  Qk+l)  desig- 
nates the  position,  in  the  transversal  plane  <r'k  (71),  of  the  point  which, 
by  GAUSS'S  Theory,  is  conjugate  to  the  point  Ql  after  refraction  at  the 
kth  surface. 

41.  Q",  Q"  are  used  to  designate  the  points  where  a  pair  of  paraxial 
rays,  of  colours  X,  X,  respectively,  both  emanating  from  the  same 
extra-axial  object-point  <2,  cross  the  focussing  plane  in  the  Image- 
Space  of  the  optical  system  (Chap.  XIII). 

In  Fig.  161,  <2"  designates  the  centre  of  the  "blur-circle"  in  the 
scale-plane  a"  of  an  optical  measuring  instrument,  which  corresponds 
to  the  extra-axial  image-point  Q1 '. 

42.  See  use  of  this  letter  in  conjunction  with  P,  R  and  5  (38). 

R 

43.  The  letter  R  is  used,  especially  in  Chap.  XIV,  to  designate  a 
point  of  a  3 -dimensional  object,  and  Rr  to  designate  the  conjugate 
point  of  the  relief-image. 

44.  See  38  for  use  of  this  letter  in  conjunction  with  P,  Q,  S. 

S 

45.  5,  5  and  S',  3'  are  used,  especially  in  the  theory  of  the  refract- 
ion of  a  narrow  bundle  of  rays,  to  designate  the  primary  and  secondary 
object-points  and  image-points,  respectively.     Thus,  S,  S'  designate  a 
pair  of  conjugate  points  on  the  chief  ray  of  a  pencil  of  meridian  rays, 
before  and  after  refraction,  respectively;  and,  similarly,  5,  S'  desig- 
nate a  pair  of  conjugate  points  on  the  chief  ray  of  a  pencil  of  sagittal 
rays,  before  and  after  refraction,  respectively.     If  the  bundle  of  inci- 
dent rays  is  homo  centric,  the  points  5,  5  coincide  at  the  vertex  of  the 
bundle. 

S'k  (or  Sk+l)  and  S'k  (or  Sk+l)  designate  the  positions  on  the  chief 
ray  of  the  primary  and  secondary  image-points,  respectively,  after 
refraction  of  the  narrow  bundle  of  rays  at  the  kth  surface  of  a  system 
of  refracting  surfaces. 

46.  5,  S'  and  5,  S'  are  used  also  as  explained  in  38  above.     5,  S' 
occur  frequently  to  designate  a  pair  of  conjugate  points  of  two  collinear 
systems. 

47.  Especially,  the  letters  5,  S'  designate  the  positions  on  the  axis 
of  the  centres  of  the  entrance-port  and  exit-port,  respectively,  of  an 
optical  system.     If  the  system  has  two  entrance-ports,  the  centres  are 
designated  by  5,  and  S2. 


592  Geometrical  Optics,  Appendix. 


48.  T,  Tf  are  used  in  Chap.  IX  to  designate  the  points  of  inter- 
section of  a  pair  of  incident  rays  lying  in  the  plane  of  a  principal  section 
of  the  spherical  refracting  surface  and  the  pair  of  corresponding  re- 
fracted rays,  respectively.     See  Fig.  121. 

49.  T  is  used  also  to  designate  the  upper  end  of  the  diameter,  in 
the  plane  of  the  principal  section,  of  the  entrance-port  of  the  optical 
system;  and  T'  designates  the  point  in  the  circumference  of  the  exit- 
port  which  is  conjugate  to  T.     If  the  system  has  two  entrance-ports, 
the  upper  ends  of  the  diameters,  in  the  plane  of  the  principal  section, 
are  designated  by  7^  and  T2. 

50.  In  certain  diagrams  of  Chap.  V,  the  letter  T  is  used  to  desig- 
nate the  infinitely  distant  point  of  the  ^-axis. 

U 

51.  This  letter  occurs  in  various  uses.     We  mention  here  only  one 
of  these,  viz. :  U  designates  the  lower  end  of  the  diameter,  in  the  plane  of 
the  principal  section,  of  the  entrance-port  of  an  optical  system ;  and  U' 
designates  the  point  in  the  circumference  of  the  exit-port  which  is  con- 
jugate to    U.     If  the  system  has  two  entrance-ports,  the  lower  ends 
of  the  diameters  are  designated  by  Uv   U2. 

V 

52.  In  the  prism-diagrams,  V  designates  the  vertex  of  the  prism. 
In  a  system  of  prisms  whose  refracting  edges  are  all  parallel,  Vk  desig- 
nates the  point  where  the  refracting  edge  of  the  &th  prism  meets  the 
plane  of  the  principal  section. 

There  are  also  various  other  uses  of  this  letter  which  it  is  not  neces- 
sary to  enumerate. 

W 

53.  This  letter  occurs  frequently  in  various  ways. 

X 

54.  This  letter  occurs  in  various  ways. 

Y 

55.  Y,  Y'  are  used  to  designate  the  feet  of  the  perpendiculars  let 
fall  from  the  centre  of  the  spherical  refracting  surface  on  the  incident 
and  refracted  rays,  respectively  (Fig.  120). 

The  letter  Y  occurs  also  in  various  other  connections. 


Designations  of  Lines.  593 

Z 

56.  In  Chap.  IX,  Z,  Z'  designate  the  points  where  the  incident 
and  refracted  rays  cross  the  auxiliary  concentric  spherical  surfaces 
T,  T'  (72),  respectively,  which  are  used  in  YOUNG'S  construction  of 
the  path  of  a  ray  refracted  at  a  spherical  surface  (Figs.  114,  115). 

In  particular,  Z,  Z'  designate  the  positions  on  the  optical  axis  of 
the  pair  of  aplanatic  points  of  a  spherical  refracting  surface  (Fig.  116). 
The  letter  Z  is  used  also  in  various  other  ways. 

II.     DESIGNATIONS    OF   LINES. 

Lines  in  the  diagrams  are  designated  generally  by  italic  small  letters. 
Without  undertaking  to  enumerate  all  the  uses  of  these  letters,  we 
may  mention  here  the  following  as  among  the  most  important. 

f,e' 

57.  In  Chapters  V,  VI  and  VII,  the  letters/  and  e'  are  frequently 
used  to  designate  the  Focal  Lines  (or  "  Flucht"  Lines)  of  two  collinear 
plane-fields;  or  the  lines  in  which  the  Focal  Planes  <£,  e'  (73,  65)  are 
intersected  by  conjugate  meridian  planes  containing  the  principal  axes 
x,  x',  respectively,  of  two  collinear  space-systems.     See  Figs.  64,  a 
and  6,  and  65. 

M 

58.  In  Chap.  VII,  the  letters  i,  i'  (and  similarly  also  the  letters 
7,  /)  are  used  to  designate  the  infinitely  distant  straight  line  and  the 
"Flucht"  Line,  respectively,  of  two  collinear  plane-fields. 

s,s' 

59.  In  Chapters  V,  VI  and  VII,  s,  s'  are  used  to  designate  a  pair 
of  conjugate  rays  of  two  collinear  systems;  as  in  Fig.  66. 

u,  uf 

60.  Throughout  Chap.  XI,  u,  u'  are  used  to  designate  the  chief 
incident  ray  and  the  corresponding  refracted  ray,  respectively,  of  an 
infinitely  narrow  bundle  of  rays  refracted  at  a  spherical  surface. 

x,  x'\  y,  y'\  2,  z' 

61.  x,  xf  designate  the  Principal  Axes  of  two  collinear  systems. 
In  an  optical  system  of  centered  spherical  surfaces,  the  optical  axis 
is  designated  by  x  or  x'  according  as  it  is  regarded  as  belonging  to 
the  Object-Space  or  Image-Space,  respectively. 

In  Chap.  VII,  xk,  x'k  designate  the  Principal  Axes  of  the  &th  com- 
ponent of  a  compound  optical  system. 

39 


594  Geometrical  Optics,  Appendix. 

62.  x,  x'\  y,  y'\  z,  z'  are  used  also  to  designate  corresponding  (but 
not  necessarily  conjugate)  pairs  of  rectangular  axes  of  co-ordinates  in 
the  Object-Space  and  Image-Space. 

63.  y  designates  the  axis  of  collineation  of  two  centrally  collinear 
plane-fields.     It  designates  especially  the  tangent-line  in  the  meridian 
plane  at  the  vertex  of  a  spherical  refracting  surface. 

64.  It  may  also  be  mentioned  that  z  is  used  (with  suitable  primes, 
subscripts,  etc.)  to  designate  the  chief  ray  of  a  narrow  bundle  of  rays 
refracted  at  the  edge  of  a  prism.     See  Fig.  42. 

III.     DESIGNATIONS   OF   SURFACES. 

Surfaces,  plane  or  curved,  are  designated  by  small  letters  of  the 
Greek  alphabet.     Of  these  the  following  are  the  more  important. 


65.  e,  e'  are  used  to  designate  the  infinitely  distant  plane  of  the 
Object-Space  and  the  Focal  Plane  of  the  Image-Space,  respectively, 
of  two  collinear  space-systems.     Similarly,  in  the  case  of  a  compound 
optical  system,  ek  designates  the  secondary  focal  plane  of  the  &th  com- 
ponent. 

"n 

66.  17,  t\  are  used  sometimes  (see  Chap.  VII)  to  designate  a  pair 
of  conjugate  plane-fields  of  two  collinear  space-systems. 

M 

67.  /*  is  used  to  designate  the  refracting  or  reflecting  surface.     If 
there  are  a  series  of  such  surfaces,  nk  designates  the  kth  surface  of 
the  series  reckoned  in  the  order  in  which  they  are  encountered  by  the 
rays  of  light. 

7T 

68.  TT,  IT'  are  used  frequently,  especially  in  Chap.  VII,  to  designate 
two  collinear  plane-fields. 

These  symbols  are  also  employed,  especially  in  Chap.  XI,  to  desig- 
nate the  coincident  planes  of  incidence  and  refraction  of  the  chief 
incident  ray  and  the  corresponding  refracted  ray,  respectively,  of  an 
infinitely  narrow  bundle  of  rays  refracted  at  a  spherical  (or  plane) 
surface.  In  particular  TT,  IT'  designate  the  collinear  plane-fields  of  the 
meridian  sections  of  a  narrow  bundle  of  incident  rays  and  the  bundle 
of  corresponding  refracted  rays. 

In  the  same  way,  also,  TT,  TT'  are  used  to  designate  the  pair  of  planes, 
both  at  right  angles  to  the  plane  of  incidence  of  the  chief  ray  of  a 


Designations  of  Surfaces.  595 

narrow  bundle  of  rays  refracted  at  a  spherical  (or  plane)  surface, 
which  contain  the  chief  incident  ray  and  the  corresponding  refracted 
ray,  respectively.  And,  especially,  TT,  TT'  designate  the  two  collinear 
plane-systems  of  the  sagittal  sections  of  the  bundles  of  incident  and 
refracted  rays,  respectively. 

Moreover,  the  symbols  ir'k,  Tr'k  designate  the  plane-systems  of  the 
meridian  and  sagittal  sections,  respectively,  after  refraction  of  a  narrow 
bundle  of  rays  at  the  kth  surface  of  a  series  of  refracting  surfaces. 

(7,  <T 

69.  cr,  cr'  are  used  in  Chap.  VII  to  designate  a  pair  of  conjugate 
planes  parallel  to  the  Focal  Planes. 

Especially,  the  symbols  o-,  a'  are  used  to  designate  a  pair  of  trans- 
versal planes  which  are  conjugate,  in  the  sense  of  GAUSS'S  Theory, 
with  respect  to  either  a  single  spherical  refracting  (or  reflecting)  surface 
or  a  centered  system  of  spherical  surfaces.  In  this  case,  a  designates 
the  so-called  Object- Plane  (Chap.  XII)  which  is  denned  as  the  trans- 
versal plane  (perpendicular  to  the  optical  axis)  which  contains  the 
object-point  P  (or  Q);  see  36,  39.  The  axial  point  M'  conjugate, 
by  GAUSS'S  Theory,  to  the  point  M  (24)  where  the  optical  axis  crosses 
the  Object-Plane  a  determines  the  position  of  the  transversal  Image- 
Plane  a' .  In  Chap.  XIV,  the  planes  v,  a'  are  usually  called  the 
Focus- Plane  and  the  Screen- Plane,  respectively. 

In  the  case  of  a  centered  system  of  spherical  surfaces,  <rk  is  used  to 
designate  the  transversal  plane  which,  by  GAUSS'S  Theory,  is  conjugate 
to  the  Object- Plane  al  with  respect  to  the  optical  system  composed 
of  the  first  k  surfaces. 

70.  a"  is  employed  to  designate  a  transversal  plane  of  the  Image- 
Space  of  an  optical  system  which  is  usually  not  far  from  the  Image- 
Plane  a'.     For  example,  in  Fig.  161,  <r"  designates  the  so-called  Scale- 
Plane  of  an  optical  measuring  instrument. 

71.  The  symbols  o-,  <r'  are  used  to  designate  a  second  pair  of  trans- 
versal planes  conjugate  to  each  other  in  the  same  way  as  o-,  a'  above. 
Generally,  <r,  <r'  designate  (as  always  in  Chapters  XII  and  XIV)  the 
planes  of  the   Entrance- Pupil  and   Exit- Pupil,  respectively,  of  the 
optical  system. 

In  the  case  of  a  centered  system  of  spherical  surfaces,  <r'k  designates 
the  transversal  plane  which,  by  GAUSS'S  Theory,  is  conjugate  to  the 
initial  plane  o^  in  the  Object-Space,  with  respect  to  the  optical  system 
composed  of  the  first  k  surfaces. 


596  Geometrical  Optics,  Appendix. 


72.  T,  T'  are  used  to  designate  the  auxiliary  spherical  surfaces, 
concentric  with  the  spherical  refracting  surface,  used  in  YOUNG'S  Con- 
struction of  the  path  of  the  refracted  ray  (Figs.  114  and  115). 


73.  <f>,  <p'  are  used  to  designate  the  Focal  Plane  (or  "  Flucht"  Plane) 
of  the  Object-Space  and  the  infinitely  distant  plane  of  the  Image- 
Space,  respectively,  of  two  collinear  space-systems.  Similarly,  in  the 
case  of  a  compound  optical  system,  (pk  designates  the  primary  focal 
plane  of  the  &th  component. 

IV.     SYMBOLS   OF   LINEAR   MAGNITUDES. 

Introduction.  A  straight  line  is  divided  into  two  segments  by  a 
pair  of  actual  (or  "finite")  points  A,  B  on  the  line,  viz.,  a  segment  of 
finite  magnitude  which  is  the  shortest  distance  between  the  two  points 
and  another  segment  of  unlimited  length  which  is  the  "long  way" 
between  the  two  points  via  the  infinitely  distant  point  I  of  the  straight 
line.  Three  actual  points  A  ,  B,  C  lying  along  a  straight  line  determine 
a  certain  "sense"  ABC  along  the  line  or  direction  in  which  the  line 
has  to  be  traversed  in  order  to  go  from  A  to  B  without  passing  through 
C.  As  we  shall  exclude  infinitely  great  line-segments,  the  segment 
A  B  is  to  be  understood  therefore  as  meaning  always  the  finite  one  of 
the  two  above-mentioned;  and  as  indicating  also  not  merely  the  dis- 
tance from  A  to  B  but  the  segment  AB  in  the  sense  ABI.  Evidently, 
therefore,  we  have  the  following  relation: 

AB  +  BA  =  o. 

Also,  if  A,  B,  C  are  three  points  ranged  along  a  straight  line  in  any 
order  whatever,  we  may  write  according  to  the  above: 

AB  +  BC  +  CA  =  o; 

and,  generally,  in  the  case  of  any  number  of  points  lying  on  one 
straight  line,  a  similar  relation  will  exist. 

If  A,  B,  C,  D,  •  •  •  designate  a  series  of  points  ranged  along  a 
straight  line,  the  segments  AB,  AC,  AD,  •  •  -  are  called  here  (for 
lack  of  a  better  term)  the  "abscissae"  of  the  points  B,  C,  ?>,  •••, 
respectively,  with  respect  to  the  point  A  as  origin. 

As  a  rule,  to  which,  however,  there  are  some  notable  exceptions 
(as  will  be  seen  in  the  following),  linear  magnitudes  are  denoted  by 
italic  small  letters.  Italic  capital  letters  and  Greek  letters  occur  some- 


Symbols  of  Linear  Magnitudes.  597 

times  as  symbols  of  linear  magnitudes.     The  more  important  of  these 
magnitudes  will  be  found  in  the  following  list. 


74.  In  Chap.  X,  the  symbol  ak  is  used  to  denote  the  abscissa  of 
the  centre  Ck+l  of  the  (k  -f  i)th  surface  with  respect  to  the  centre 
Ck  of  the  kth  surface  of  a  centered  system  of  spherical  surfaces;  thus 

^k   =   ^Jt^k+r 

b,  B 

75.  In  Chap.  I  X,  b,  b'  denote  the  intercepts  of  a  ray,  lying  in  the 
principal  section  of  a  spherical  refracting  surface,  on  the  central  per- 
pendicular, before  and  after  refraction,  respectively;  thus  b  =  CH, 
b'  =  CH'  (6  and  15).     Similarly,  in  Chap.  X,  bk  =  CkHk,  b'k  =  CkH'k. 

76.  In  Chap.  XIII,  6,  b'  are  used  to  denote  the  widths  of  a  pencil 
of  parallel  meridian  rays  before  and  after  refraction,  respectively,  at 
a  plane  surface.     Similarly,  bk  denotes  the  width  of  a  pencil  of  parallel 
meridian  rays  after  refraction  at  the  kth  surface  of  a  system  of  prisms 
with  their  refracting  edges  all  parallel. 


77.  In  Chap.  IX,  c,  c'  denote  the  abscissae,  with  respect  to  the 
centre  C  of  the  spherical  refracting  surface,  of  the  points  designated 
by  L,  L'  (23);  thus,  c  =  CL,  c'  =  CL'. 

78.  In  Chap.  XIV,  c,  c'  denote  the  abscissae,  with  respect  to  the 
centres  of  the  pupils,  of  the  centres  of  the  ports  (27  and  47);  thus, 
c  =  MS,  c'  =  M'S'.    Also,  c,  =  MSV  c2  =  MS2. 


79.  d  denotes  the  axial  thickness  of  an  optical  medium  comprised 
between  two  consecutive  surfaces  of  a  centered  system  of  spherical 
surfaces.     Particularly,  dk  =  AkAk+l  (see  2). 

In  an  optical  system  composed  of  a  single  lens,  the  thickness  of  the 
lens  is  denoted  by  d;  thus,  d  =  A±A2. 

80.  In  a  centered  System  of  Infinitely  Thin  Lenses,  dk  denotes  the 
distance  of  the  (k  +  i)th  lens  from  the  kth  lens;  thus,  dk  =  AkAk+l 
(see  2). 

81.  ^  In  Chap.  VIII,  in  an  optical  system  consisting  of  a  combina- 
tion of  two  lenses,  d  is  used  to  denote  the  abscissa,  with  respect  to 
the  secondary  principal  point  of  the  first  lens,  of  the  primary  prin- 
cipal point  of  the  second  lens;  thus,  d  =  A(A2  (see  i). 

82.  The  symbol  5,.  is  employed  to  denote  the  length  of  the  ray-path 
comprised  between  the  kth  and  the  (k  +  i)th  refracting  surfaces;  thus, 

i  (see  5)- 


598  Geometrical  Optics,  Appendix. 

83.  Here  also  we  note  the  use  of  the  symbol  Ak  to  denote  the  so- 
called  "optical  interval11  between  the  &th  and  the  (k  -j-  i)th  components 
of  a  compound  optical  system;  thus,  AA  =  E'kFk+l  (see  n  and  12). 
If  the  compound  system  has  only  two  parts,  we  write:  A  =  E{F2. 


84.  The  secondary  focal  length  of  an  optical  system  is  denoted  by 
e'\  that  is,  er  =  E'A'  (i  and  1 1).     Also,  in  a  compound  optical  system, 
4  =  E'kA'k.     Also,  in  Chap.  XIII,  e'  =  E'A'. 

85.  In  the  theory  of  the  refraction  of  a  narrow  bundle  of  rays  at 
a  spherical  surface  (or  through  a  centered  system  of  spherical  surfaces) , 
the  symbols  e'u  and  eu  are  used  in  Chap.  XI  to  denote  the  secondary 
focal  lengths  of  the  two  collinear  plane-systems  TT,  IT'  and  TT,  TT',  re- 
spectively (68).     The  subscript  u  refers  to  the  chief  ray  of  the  bundle 
of  incident  rays  (60). 

Similarly,  the  secondary  focal  lengths  of  the  systems  of  meridian 
and  sagittal  rays  of  an  infinitely  narrow  bundle  of  rays  which  are 
refracted  at  the  &th  surface  of  a  centered  system  of  spherical  surfaces 
are  denoted  by  e'Ut  *,  eUf  &,  where  u  designates  the  chief  ray  of  the  bundle 
of  object-rays. 

/ 

86.  The  primary  focal  length  of  an  optical  system  is  denoted  by 
/;  thus,/  =  FA  (see  I  and  12).     Also,  in  a  compound  optical  system, 
fk  =  FkAk.    Also,  J=  FA  (see  Chap.  XIII). 

87.  The  symbols  /M  and  fu  are  used  in  the  same  connection  as  eu 
and  eu  (85)  to  denote  the  primary  focal  lengths  of  the  systems  TT,  TT' 
and  TT,  TT',  respectively.     Similarly  also  the  symbols  /Mi  kJ  JUt  k,  corre- 
sponding to  eut  k,  e'Uj  k,  respectively. 

t   „  & 

88.  The  symbols  g,  "g  are  used  in  Chap.  XIII  to  denote  the  ordi- 
nates  of  the  points  where  an  incident  paraxial  ray  emanating  from  the 
axial  object-point  M  crosses  the  primary  focal  planes  of  an  optical 
system  which  correspond  to  light  of  wave-lengths  X,  X,  respectively. 
Also,  in  Chapters  V,  VI  and  VII,  the  symbol  g  is  employed  in  a  sense 
similar  to  the  above.     See  Figs.  65  and  90,  where  g  =  FR. 

h,h 

89.  The  symbol  h  is  used  to  denote  the  incidence-height  (or  ordinate 
of  the  incidence-point  B)  of  a  ray  refracted  (or  reflected)  at  a  spherical 
surface;  thus,  h  =  DB  (5  and  8).     With  respect  to  a  centered  system 
of  spherical  surfaces,  hk  =  DkBk  denotes  the  incidence-height  at  the 


Symbols  of  Linear  Magnitudes.  599 

kth  surface  of  a  ray  lying  in  the  principal  section.     In  Chap.  XIII, 
we  have  also  hk  =  DkBk. 

90.  Similarly,  hk  =  DJSk  (5  and  8)  denotes  the  incidence-height 
of  a  second  ray,  usually  the  chief  ray,  at  the  kth  surface  of  a  centered 
system  of  spherical  surfaces. 

91.  The  symbol  h  is  used  to  denote  the  incidence-height  of  a  ray 
refracted  through  an  Infinitely  Thin  Lens.     In  a  centered  system  of 
Infinitely  Thin  Lenses,  hk  denotes  the  incidence-height  at  the  kth  lens. 

92.  In  the  case  of  two  collinear   space-systems  (Chap.  VII),  the 
symbols  h,  h'  are  used  to  denote  the  ordinates  of  the  points  where  a 
pair  of  conjugate  rays  cross  the  primary  and  secondary  principal  planes, 
respectively;  h'  =  h.     In  a  compound  optical  system,  hk)  hk  (=  hk) 
are  used  in  this  same  sense  with  respect  to  the  kth  component;  see 
Fig.  99. 

k 

93.  The  symbol  k'  is  used,  always  in  connection  with  the  symbol 
g  (88),  to  denote  the  ordinate  of  the  point  where  a  paraxial  image-ray 
lying  in  the  plane  of  the  principal  section  crosses  the  secondary  focal 
plane;  see  Fig.  65. 

I 

94.  The  symbols  /,  /'  are  used  to  denote  the  so-called  "ray-lengths" 
of  a  ray  lying  in  the  principal  section  of  a  spherical  refracting  surface, 
before  and  after  refraction,  respectively;  reckoned  in  each  case  from 
the  incidence-point  B  to  the  point  where  the  ray  crosses  the  optical 
axis;   thus,  /  =  BL,  I'  =  BL'  (23).     In  the  case  of  a  ray  lying  in 
the  princiapl  section  of  a  centered  system  of  spherical  refracting  surf- 
aces, lk  =  BkL'fe_l,  l'k  =  BkL'k  denote  the  ray-lengths,  before  and  after 
refraction,  respectively,  at  the  kth  surface. 

P 

95.  In  Chap.  I X,  p,  pr  denote  the  radii  vectores  of  the  points  H,  H' 
(15);  thus  p  =  CH,  p'  =  CHf  (Fig.  123).     In  Chap.  X,  pk  =  CkHk, 
P'k=  CkH'k. 

96.  In  Chap.  XIV,  p,  pr  denote  a  pair  of  conjugate  radii  of  the 
Entrance- Pupil  and  Exit- Pupil,  respectively,  of  the  optical  system; 
thus,  p  =  MD,  p'  =  M'D'  (9  and  27). 

The  symbol  pQ  occurs  to  denote  the  radius  of  the  iris-opening  of 
the  eye. 

a 

97.  In  Chap.  XIV,  g,  g'  denote  a  pair  of  conjugate  radii  of  the 


600  Geometrical  Optics,  Appendix. 

Entrance- Port  and  Exit- Port,  respectively,  of  the  optical  system;  thus, 
2  =  ST,  qf  =  ST  (47  and  49).     Also,  ql  =  S^,  g,  =  S2T2. 


98.  The  symbol  r  is  used  to  denote  the  radius  of  the  spherical 
refracting  surface ;  or,  more  exactly,  to  denote  the  abscissa  of  the  centre 
C  with  respect  to  the  vertex  A ;  r  —  A  C.     Similarly,  rk  =  AkCk  (2 
and  6)  denotes  the  radius  of  the  &th  surface. 

99.  The  symbols  R,  R  and  R',  R'  are  used  to  denote  the  radii  of 
curvature  at  the  axial  points  M  and  M'  of  the  I.  and  II.  image-surfaces, 
before  and  after  refraction,  respectively,  at  a  spherical  surface  (Chap. 
XII);  R  =  MK,    R'  =  M'K',   R  =  MK,  R'  =  M'K'  (22  and  24). 
Also,  R'k,  Rk  are  used  in  the  same  way,  with  respect  to  the  astigmatic 
image-surfaces  after  refraction  at  the  fcth  spherical  surface. 


100.  The  symbols  5,  s'  are  used  to  denote  the  distances,  reckoned 
in  each  case  from  the  incidence-point  B  of  the  chief  ray,  of  the  vertex 
S  of  an  infinitely  narrow  pencil  of  meridian  rays  and  the  vertex  5' 
of  the  pencil  of  corresponding  refracted  rays,  respectively;  thus,  5  =  BS, 
s'  =  BS'  (5  and  45).  Similarly  s,  s'  denote  the  distances,  from  the 
incidence-point  B  of  the  chief  ray,  of  the  vertex  »S  of  an  infinitely  nar- 
row pencil  of  sagittal  rays  and  the  vertex  S'  of  the  pencil  of  corre- 
sponding refracted  rays,  respectively;  J  =  B15,  "sf  =  B5f  (5  and  45). 

If  the  rays  traverse  a  system  of  prisms  or  a  centered  system  of 
spherical  surfaces,  we  have  with  respect  to  the  &th  surface: 


ii     sk  =  BkSk  =  BkSk+l. 

t 

101.  In  Chap.  IX,  /,  /'  denote  the  distances  from  the  incidence- 
point  B  of  a  chief  ray  lying  in  the  plane  of  the  principal  section  of  a 
spherical  surface  of  the  points  T,  T'  of  intersection  with  this  ray  of 
another  meridian  ray,  before  and  after  refraction,  respectively;  thus, 
/  =  BT,  t'  =  BT',  as  in  Fig.  121.     See  5  and  48. 

«,  u,  U 

102.  The  symbols  «,  u'  are  used  to  denote  the  abscissae,  with  respect 
to  the  principal  points  A,  A'  of  two  collinear  systems,  of  a  pair  of 
conjugate  axial  points  Af,  M'  respectively;  thus,  u  =  AM,  u'  =  AM' 
(2  and  24). 


Symbols  of  Linear  Magnitudes.  601 

103.  Especially,  u,  u'  denote  the  abscissae,  with  respect  to  the 
vertex  A  of  the  spherical  surface,  of  the  points  M,  M'  where  a  paraxial 
ray  crosses  the  optical  axis,  before  and  after  refraction  (or  reflexion), 
respectively;  u  =  AM,  u'  =  AM'  (2  and  24,  25  ). 

If  (as  in  Chap.  VIII,  §  195)  we  have  a  pair  of  paraxial  rays  of  dif- 
ferent origins,  the  abscissae  of  the  points  M,  M'  where  the  second  ray 
crosses  the  axis  before  and  after  refraction  are  denoted  by  u,  u', 
respectively;  u  =  AM,  u'  =  AM'  (2  and  26). 

In  case  we  are  concerned  with  paraxial  rays  of  two  different  colours 
emanating  from  a  common  source,  the  symbols  u,  u'  and  u,  u'  are  used 
as  above  described  with  reference  to  rays  of  light  of  wave-lengths  X,  X, 
respectively;  and  if  also  there  is  a  ray  of  a  third  colour  I,  the  abscissae 
of  the  points  where  this  ray  crosses  the  axis  are  denoted  by  tt,  \tf 
(see  Chap.  XIII). 

In  an  optical  system  consisting  of  a  centered  system  of  spherical 
surfaces,  the  symbols  uk,  uk\  uk,  uk\  ukl  uk\  etc.,  are  used  precisely  in 
the  same  way  as  described  above,  with  respect  to  the  kth  surface  of 
the  system;  so  that 

uk  =  AkMk  =  AkMk_lt    uk  =  AkM'k  =  AkMk+l,  etc. 

In  particular,  the  symbol  u±  =  AlMl  denotes  the  abscissa,  with 
respect  to  the  vertex  Ap  of  the  axial  object-point  Mr  Frequently, 
however,  the  symbol  u,  without  any  addition,  is  used  to  denote  the 
abscissa  of  the  point  where  a  paraxial  object-ray  crosses  the  optical 
axis  of  a  centered  system  of  spherical  surfaces;  in  which  case  u'  de- 
notes the  abscissa,  with  respect  to  the  vertex  of  the  last  surface,  of 
the  point  where  the  conjugate  image-ray  crosses  the  axis. 

104.  In  the  case  of  an  Infinitely  Thin  Lens,  the  symbols  u,  uf 
denote  the  abscissae,  with  respect  to  the  optical  centre  of  the  lens,  of 
the  points  where  a  paraxial  ray  crosses  the  axis  before  entering  the 
lens  and  after  emerging  from  it,  respectively.     The  symbols  u,  u'  and 
Uj  u'  are  used  also  in  this  way. 

Similarly,  in  the  case  of  a  centered  system  of  Infinitely  Thin  Lenses, 
the  symbols  uk,  uk  are  used  as  just  stated,  with  respect  to  the  kth  lens. 
So  also  «£,  uk  and  uk,  uk. 

105.  In  general,  the  symbol  ul  —  A^M^  denotes  the  abscissa,  with 
respect  to  the  vertex  Al  of  the  centre  Ml  of  the  Entrance- Pupil  of 
the  system.     If  the  centres  of  the  Entrance-Pupil  and  Exit-Pupil  are 
designated  by  M,  M'  (27),  then  u  =  AvMj  u'  =  AmM'  are  used  to 
denote  the  abscissa  of  the  pupil-centres. 


602  Geometrical  Optics,  Appendix. 


v,  V 

106.  The  symbols  v,  v'  denote  the  abscissae,  with  respect  to  the 
vertex  A  of  the  spherical  refracting  surface,  of  the  points  L,  L'  (23) 
where  a  ray  lying  in  the  plane  of  the  principal  section  crosses  the  optical 
axis,  before  and  after  refraction,  respectively;  v  =  AL,  v'  =  AL'. 
Also:  vk  =  AkLk  =  AkL'k_^  vk  =  AkL'k  =  AkLk+l. 

The  symbols  v,  v'  have  meanings  with  respect  to  the  chief  ray  pre- 
cisely the  same  as  above;  thus,  v  =  AL,  v'  =  AL';  also,  vk  =  AkL'k_lt 
»I  =  A£k  (see  23). 

In  Chap.  IX,  in  KERBER'S  formulae  for  the  path  of  an  oblique  ray 
refracted  at  a  spherical  surface,  we  have: 


where  the  points  designated  by  Ag,  A^  G,  G',  I  and  /'  are  points 
described  in  3,  14  and  18.  See  Fig.  122.  Also,  in  Chap.  X,  in  the 
same  connection  we  have: 


X,   X 

107.  x,  xr  denote  especially  the  abscissae,  with  respect  to  the  focal 
points  F,  E',  of  a  pair  of  conjugate  axial  points  M,  M',  respectively, 
of  two  collinear  systems;  thus,  x  =  FM,  x'  =  E'M'.     Similarly,  with 
reference  to  the  &th  component  of  a  compound  optical  system,  we 
have:  xk  =  FkM'k,  xk  =  E'kM'k  (n,  12  and  24). 

In  Chap.   XIII,  x,  x'  occur  in  connection  with  the  Focal  Points 
?,  Ef  (u  and  12). 

In  Chap.  VII,  the  letters,  x,  x'  occur  also  with  special  subscripts. 

108.  jc,  *'  denote  the  abscissae,  with  respect  to  the  focal  points 
F,  E',  of  the  pupil-centres;  thus,  x  =  FM,  x'  =  E'M'  (n,  12  and  27). 

109.  The  letters  x,  y,  z  and  x',  yf,  z'  are  used  to  denote  the  rectangu- 
lar co-ordinates  of  a  pair  of  conjugate  points  of  two  collinear  space- 
systems. 

110.  In  Chap.  IX,  xg,  xg  and  x^  x\  denote  the  ^-co-ordinates,  with 
respect  to  the  centre  C  of  the  spherical  refracting  surface,  of  the  points 
designated  by  G,  Gr  and  7,  /',  respectively;  and,  similarly,  in  Chap. 
X,  Xgt  fc,  x'gt  &,  and  #,-,  *,  x\t  *  denote  the  x-co-ordi  nates,  with  respect  to  Ck, 
of  the  points  Gk,  Gk  (or  Gk+l)  and  Ik,  I'k  (or  Ik+l)y  respectively  (6,  14 
and  1  8). 


Symbols  of  Linear  Magnitudes.  603 

y,  y 

111.  y,  y'  denote  the  ^-co-ordinates  of  a  pair  of  conjugate  points 
of  two  collinear  space-systems;  especially,  the  ordinates  of  the  extra- 
axial  conjugate  points  Q,  Q'  lying  in  the  meridian  ^-plane;  y  =  MQ, 
y'  =  M'Q'  (24  and  39). 

y'  denotes  the  ordinate  of  the  vertex,  after  refraction  at  the  kth 
surface  of  a  centered  system  of  spherical  surfaces,  of  a  bundle  of 
paraxial  rays  which  emanate  originally  from  the  extra-axial  object- 
point  Ql  lying  in  the  plane  of  the  principal  section ;  or  the  ordinate  of 
the  point  Q'k  (or  Qk+i),  where,  according  to  GAUSS'S  Theory,  a  ray 
emanating  from  the  object-point  Ql  (or  PJ  would  cross  the  trans- 
versal plane  ah  after  refraction  at  the  kth  surface  of  a  centered  system 
of  spherical  surfaces.  See  36,  39  and  69.  Thus,  y'k  =  M'kQ'k. 

In  Chap.  XIII,  where  we  have  to  do  with  rays  of  light  of  two  or 
more  different  colours,  the  symbols  y,  y'  denote  the  ordinates  of  the 
pair  of  extra-axial  conjugate  points  Q,  Q'  for  rays  of  wave-length  X; 
y  =  MQ,  y'  =  M'Q'  (24  and  39);  usually,  y  =  y.  In  the  same  way, 

yk  -  3&» 

112.  y,  y'  denote  the  y-co-ordinates  of  the  points  Q,  Q',  respect- 
ively (40).     The  symbol  yk  denotes  the  y-co-ordinate   of   the   point 
Q;  (see  Chap.  XII). 

113.  In  Chap.  IX,  yg,  yg  and  yh,  y'h  denote  the  ^-co-ordinates  of 
the  points  designated  by  G,  G'  and  H,  H',  respectively  (Figs.  122  and 
123);  see  also  Chap.  XII.    In  Chap.  X,  ygtltj  yg^  and  yhtJk,  yhtJe  denote 
the  y-co-ordinates  of  the  points  designated  by  Gk,  G'k  (or  Gk+l)  and  Hk, 
H'ki  respectively  (14  and  15). 

z,  z 

114.  The  symbols  z,  z'  denote  the  z-co-ordinates  of  a  pair  of  con- 
jugate points  of  two  collinear  space-systems. 

115.  Especially  in  Chap.   XII,  z,  z'  denote  the  z-co-ordinates  of 
the  points  Q,  Q'.     If  the  object-point  Q  lies  in  the  meridian  :ry-plane, 
z  =  z'  —  o.     zk  denotes  the  z-co-ordinate  of  the  point  Q'k  (39). 

116.  z,  z',  z'k  denote  the  z-co-ordinates  of  the  points  designated 
by  Q,  Q',  Q1-,  respectively;  see  40. 

117.  In  Chap.  IX,  z/(,  zh  and  z{,  z\  denote  the  z-co-ordinates  of 
the  points  designated  by  H,  H'  and  /,  /',  respectively  (Figs.  122  and 
123).    Also,  in  Chap.  X,  zht  k,  z'ht  *  and  z<tft»  Zi,  *  denote  the  z-co-ordinates 
of  the  points  Hk,  H'k  and  IkJ  I'k,  respectively  (15  and  18). 

118.  Finally,  in  Chap.  XIII,  the  symbols  z,  z'  are  used  in  a  special 
sense  to  denote  the  abscissae,  with  respect  to  the  vertices  of  the  first 


604  Geometrical  Optics,  Appendix. 

and  last  surfaces,  of  the  primary  and  secondary  focal  points,  respect- 
ively, for  rays  of  light  of  wave-length  X;  so  that  z  =  A1F,  z'  =  AmE'. 
Similarly,  for  rays  of  light  of  wave-length  X,  we  have:  z  =  A^F, 
~zr  =  AmE'  (n  and  12). 

£»  17,  f ;  11,  I 

119.  IB  Chap.  XIV,  the  Greek  letters  £,  £'  are  used  to  denote  the 
abscissae,  with  respect  to  the  centres  of  the  entrance-pupil  and  exit- 
pupil,  of  the  pair  of  conjugate  axial  points  M,  M',  respectively;  thus, 
£  =  MM,  ?  =  M'M'  (24  and  27). 

120.  In  Chap.    XII,  the  rectangular  co-ordinates  of  the  points 
designated  by  P  and  P'  are  denoted  by  £,  17,  f  and  £',  77',  £',  respect- 
ively.    Also,  the  co-ordinates  of  P'k  are  (£,  ij'k,  $'k  (36). 

121.  In  Chap.  XII,  T|,  t\f  and  £,  £'  are  used  to  denote  the  y-  and 
z-co-ordinates  of  the  points  designated  by  P,  P' ,  respectively;  simi- 
larly, r\k,  5^.  are  used  with  reference  to  the  point  P'k  (37). 

V.     SYMBOLS   OF   ANGULAR   MAGNITUDES. 

If  A,  B,  C  designate  the  positions  of  three  points  not  in  a  straight 
line,  the  /.ABC  is  the  angle  through  which  the  straight  line  AB 
must  be  turned  in  order  that  the  point  A  may  be  brought  to  lie  in  the 
same  direction  from  the  turning-point  B  as  the  point  C  is;  thus, 
/.ABC  +  /.  CBA  =  o. 

Throughout  this  volume,  counter-clockwise  rotation  is  reckoned  al- 
ways as  positive  rotation. 

With  rare  exceptions,  angular  magnitudes  are  denoted  by  the  letters 
of  the  Greek  alphabet.  The  more  important  of  these  angles  are 
enumerated  in  the  following  list. 

a,  a,  A 

122.  The  angles  of  incidence  and  refraction,  as  denned  in  Chap.  II 
(see  Fig.  5),  are  denoted  by  a,  a',  respectively.     When  a  ray  of  light 
traverses  a  series  of  optically  isotropic  media,  the  symbols  ak,  ak  denote 
the  angles  of  incidence  and  refraction,  respectively,  at  the  kth  refracting 
surface. 

123.  The  capital  Greek  letter  A  denotes  the  critical  angle  of  inci- 
dence of  a  ray  refracted  into  a  less  dense  medium,  and  A'  denotes  the 
critical  angle  of  refraction  of  a  ray  refracted  into  a  more  dense  medium, 

124.  The  angles  of  incidence  and  refraction  at  a  spherical  surface 
of  the  so-called  chief  ray  are  denoted  by  a,  a',  respectively.    Similarly, 
with  respect  to  the  &th  refracting  surface  of  a  series  of  such  surfaces, 
the  symbols  a&,  ak  are  employed. 


Symbols  of  Angular  Magnitudes.  605 

ft 

125.  The  refracting  angle  of  a  prism  is  denoted  by  /3.     In  a  train 
of  prisms,  fik  =  Z  Vk_lVkVk+l  denotes  the  refracting  angle  of  the  kth 
prism  (52). 

d 

126.  In  KERBER'S  Refraction-Formulae  (Chap.  IX),  5,  6'  are  used 
to  denote  a  certain  pair  of  auxiliary  angular  magnitudes  relating  to 
the  ray  before  and  after  refraction,  respectively  (Fig.  122).     In  Chap. 
X,  d'f.  is  employed  in  the  same  way  with   reference  to  the  ray  after 
refraction  at  the  kth  surface  of  a  centered  system  of  spherical  surfaces. 


127.  The  acute  angle  through  which  the  refracted  ray  has  to  be 
turned  in  order  to  bring  it  into  coincidence  with  the  corresponding 
incident  ray,  the  so-called  angle  of  deviation,  is  denoted  by  c;  in  Fig.  9 
Z-P'BP  =  e.     Thus,  also,  ek  denotes  the  angle  of  deviation  at  the 
kth  refracting  surface.     The  total  deviation  of  a  ray  after  traversing  a 
train  of  prisms  with  their  edges  all  parallel  is  denoted  by  e  =  £*=7  ek, 
where  m  denotes  the  total  number  of  refracting  planes. 

The  angle  of  minimum  deviation  of  a  prism  or  prism-system  is  de- 
noted by  €0. 

128.  In  KERBER'S  Refraction-Formulae  (Chap.  IX),  e,  *'  are  used 
to  denote  a  certain  pair  of  auxiliary  angular  magnitudes  relating  to 
the  ray  before  and  after  refraction,  respectively  (Fig.   122).     Also, 
in  Chap.  X,  in  the  same  connection,  e/.  has  reference  to  the  ray  after 
refraction  at  the  kth  surface  of  a  centered  system  of  spherical  surfaces. 


6,  0  and  6,  0 

129.  0  =  Z.AMB  or  Z.ALB,  e'  =  /.AM'B  or  /.AL'B,  where 
the  points  designated  by  A,  B,  M,  M',  L,  L'  have  the  meanings 
explained  in  2,  5,  23  and  24.,  Also,  6k  =  ek+l  =  £AkM'kBk  or  Z.AkL'kBk. 
The  angles  B,  6'  are  the  so-called  slope-angles  of  the  ray  before  and 
after  refraction,  respectively,  at  a  spherical  surface. 

If  we  have  a  pair  of  rays  of  two  different  colours  emanating  origi- 
nally from  the  same  point  on  the  optical  axis  of  a  centered  system, 
0&,  ffk  denote  the  slope-angles,  after  refraction  at  the  kth  surface,  of 
the  rays  of  wave-lengths  X,  X,  respectively. 

The  symbol  0  is  used  to  denote  the  slope-angle  of  an  object-ray 
proceeding  from  the  axial  object-point  M  (24  and  25)  and  the  symbol 
6'  to  denote  the  slope-angle  of  the  conjugate  image-ray,  especially 


606  Geometrical  Optics,  Appendix. 

on  the  assumption  of  collinear  correspondence  between  Object-Space 
and  Image-Space. 

130.  The  symbols  0,  0'  are  used  to  denote  the  slope-angles  of  the 
chief  ray,  before  and  after  refraction,  respectively,  at  a  spherical  surf- 
ace; thus,  8  =  LALB,  0'  =  AAL'B.  Similarly, 


See  2.  5  and  23. 

0,  0'  denote  the  slope-angles  of  a  chief  object-ray  and  its  conjugate 
image-ray,  respectively;  especially,  on  the  assumption  of  collinear 
correspondence  between  Object-Space  and  Image-Space. 

131.  In  Chap.   XIV,  9,  9'  are  used  to  denote  the  semi-angular 
diameters  of  the  aperture  of  the  optical  system  in  the  Object-Space 
and  Image-Space,  respectively;  9  =  Z.MMD,  9'  =  Z.M'M'D'  (9,  24, 
25,  26). 

In  this  same  chapter,  9',  9o  are  used  to  denote  the  angles  subtended 
at  the  centre  of  the  image  on  the  retina  of  the  eye  by  the  radius  of 
the  exit-pupil  of  the  instrument  and  the  radius  of  the  eye-pupil,  re- 
spectively. 

132.  In  Chap.  XIV,  0,  0'  denote  the  semi-angular  diameters  of 
the  field  of  view  of  the  object  and  image,  respectively  ;  thus, 


0  =  zsMr,   0'  =  zs'M'r  (26,  47,  49). 

133.     Finally,  in  connection  with  KERBER'S  Refraction-Formulae 
(Chap.  IX),  we  have: 


where  the  points  designated  by  Ag,  At,  B,  G,  Gf  and  7,  I'  are  the 
points  explained  in  3,  5,  14  and  18.     See  Fig.  122. 
Similarly,  in  Chap.  X, 

e'g>  k  =  Z  Agt  *G'J(Bk,     0;,  k  =  Z  A  ,.,  J'kBk. 

X 

134.  In  Chap.  IX,  X,  X'  are  used  to  denote  the  angles  between  a 
pair  of  meridian  incident  rays  and  the  pair  of  corresponding  refracted 
rays,  respectively;  see  Fig.  121. 

Especially,  in  Chaps.  XI  and  XII,  the  symbols  dX,  d\'  are  used 
to  denote  the  angular  apertures  of  an  infinitely  narrow  pencil  of  merid- 
ian incident  rays  and  the  pencil  of  corresponding  refracted  rays,  re- 
spectively; thus,  d\  =  Z.BSG,  d\f  =  /.BS'G  (see  5,  13  and  45),  for 
example,  in  Fig.  127. 


Symbols  of  Angular  Magnitudes.  607 

Similarly,  d\,  d\'  denote  the  angular  apertures  of  a  narrow  pencil 
of  sagittal  incident  rays  and  the  pencil  of  corresponding  refracted 
rays,  respectively. 

The  symbols  d\'k,  d\'k  are  employed  in  the  same  way  as  above,  with 
respect  to  the  &th  surface  of  a  centered  optical  system. 


135.  In  SEIDEL'S  Refraction-Formulae,  the  symbols  /z,  ju'  are  em- 
ployed to  denote  a  pair  of  auxiliary  angles,  viz.,  the  angles  at  H",  Hf 
of  the  triangles  BHC,  BH'C,  respectively  (5,  6  and  15).  For  the 
exact  definitions  of  these  angles,  see  Chap.  IX.  In  Chap.  X,  the 
symbols  pk,  p'k  are  used  in  the  same  sense. 


136.  In  Chap.  IX,  in  SEIDEL'S  Refraction-Formulae,  ?r,  TT'  denote 
the  polar  angles  of  the  points  H,  H',  respectively;  thus,  IT  =  Z.  HCy, 
TT'  =  L  H'  Cy,  where  y  designates  a  point  on  the  positive  half  of  the 
3>-axis  of  co-ordinates  and  C,  H,  H'  have  the  meanings  given  in  6 
and  15.  See  Fig.  123.  Similarly,  also,  in  Chap.  X: 


137.  In  Chap.  I  X,  in  SEIDEL'S  Refraction-Formulae,  T,  rf  are  em- 
ployed to  denote  the  positive  acute  angles  between  the  direction  of 
the  optical  axis  (#-axis)  and  the  path  of  an  oblique  ray,  before  and 
after  refraction,  respectively,  at  a  spherical  surface  (Fig.  123).  Simi- 
larly, in  Chap.  X,  the  symbols  rk,  rk  are  used. 


138.  0  is  used  to  denote  the  central  angle  subtended  at  the  centre 
C  of  the  spherical  refracting  (or  reflecting)  surface  by  the  arc  BC\ 
thus,  0  =  Z.BCA.     Also,  0ft  =  Z  BkCkA*  (2,  5  and  6). 

139.  Similarly,  <|>  =  LBCA  or  <|>A  =  Z.BkCkAk  denotes  the  central 
angle  with  respect  to  the  so-called  chief  ray  (Fig.  121).     See  2,  5,  6. 

140.  In  Chap.  IX,  in  KERBER'S  Refraction-Formulae,  we  have: 
<f)ff  =  LA  CAg,  0(.  =  Z  A  CAv  where  the  letters  A,  A(J,  A{  and  C  have 
the  meanings  given  in  2,  3  and  6.     See  Fig.  122.     Similarly,  in  Chap. 
X:  <j>9tk  =  Z.AkCkAgti,  <j>i,k  =  ^AkCkAi,k. 

141.  In  Chap.   XIV,  <£,  <£'  are  used  in  the  radiation-formulae  to 
denote  the  angles  of  emission  and  radiation,  respectively. 


608  Geometrical  Optics,  Appendix. 

X 

142.  In  Chap.  I  X,  the  symbol  x  is  used  to  denote  the  angle  B  CB, 
where  B,  B  designate  the  incidence-points  on  a  spherical  refracting 
surface  of  a  pair  of  meridian  rays.     See  Fig.  121. 

t 

143.  In  Chap.  IX,  in  SEIDEL'S  Refraction-Formulae,  $,  \[/'  denote 
a  certain  pair  of  angular  magnitudes  (see  Fig.  123);  also,  in  Chap.  X, 
\j/k,  $'k  are  used  in  same  sense. 

VI.     SYMBOLS     OF     NON-GEOMETRICAL     MAGNITUDES     (CONSTANTS, 
CO-EFFICIENTS,    FUNCTIONS,    ETC.). 

Among  the  more  important  magnitudes  under  this  head  may  be 
mentioned  the  following: 

A 

144.  The  numerical  aperture  of  the  optical  system,  in  the  Object- 
Space  and  in  the  Image-Space,  is  denoted  by  A,  A',  respectively; 
thus,  A  =  W  -sin  6,  A'  =  w'-sin0'  (131,  155). 

B 

145.  B  =  na  =  hJ  denotes  the  optical  invariant  in  the  case  of 
paraxial  rays  (89,  122,  150,  155). 

c,  C 

146.  The  symbols  c,  c'  (sometimes  also  cv  c.2)  are  used  to  denote 
the  curvatures  of  the  surfaces  of  an  Infinitely  Thin  Lens  ;  thus, 

c  =  i/rlt     c'  =  i/r2  (see  98). 

In  a  centered  system  of  infinitely  thin  lenses,  the  symbols  ck,  c'k  denote 
the  curvatures  of  the  &th  lens.     Moreover,  in  Chap.  XIII,  Ck  =  ck  —  ck. 

147.  In  Chap.  XIV,   C,   C'  denote  the  candle-powers  of  a  point- 
source  of  light  in  a  given  direction  and  the  corresponding  point  of 
the  image  in  the  conjugate  direction,  respectively. 


148.  I  denotes  the  invariant  of  refraction  in  the  case  of  the  refract- 
ion at  a  spherical  surface  of  a  ray  of  finite  slope  lying  in  the  principal 
section:  /  =  n(v  —  r)/rl  =  n'(v'  —  r)/rlf  ;  see  94,  98,  106  and  155. 

149.  In  Chap.  XIV,  i,  if  denote  the  specific  intensities  of  radiation 
of  a  luminous  surface-element  in  a  given  direction  and  the  corre- 
sponding element  of  the  image  in  the  conjugate  direction,  respectively. 


Symbols  of  Non-Geometrical  Magnitudes.  609 

J,J 

150.  The  symbols  J,  J  denote  the  so-called  "zero-invariants"  in 
the  case  of  the  refraction  of  paraxial  rays  at  a  spherical  surface,  with 
respect  to  the  two  pairs  of  conjugate  axial  points  M,  M'  and  M,  M', 
respectively  (24,  25,  26  and  27);  thus, 


(see  98,  103,  155). 

In  the  case  of  a  centered  system  of  spherical  surfaces,  Jk,  Jk  denote 
the  zero-invariants  for  the  &th  surface,  with  respect  to  the  pairs  of 
conjugate  axial  points  Mk_v  M'k  and  Mk_v  Mk,  respectively. 

In  Chap.  XIII,  JkJ  Jk  denote  the  zero-invariants,  with  respect  to 
the  &th  surface,  for  paraxial  rays  of  light  of  colours  X,  X,  respectively, 
emanating  originally  from  the  same  axial  object-point. 

K,  k 

151.  K  =  n-sin  a  =  n'  -sin  a.'  denotes  the  magnitude  of  the  optical 
invariant  in  the  refraction  of  a  ray  of  light  (122  and  155). 

152.  The  symbol  k,  which  occurs  usually  as  a  subscript,  denotes 
the  series-number  of  any  one  of  a  system  of  refracting  (or  reflecting) 
surfaces;  or  of  any  integral  part  or  component  of  a  compound  optical 
system.     In  certain  prism-formulae,  the  subscripts  i  and  r  occur  also 
in  this  same  sense. 

L 

153.  In  Chap.   XIV,  L,  L'  are  used  to  denote  the  quantities  of 
light-energy  emitted  in  unit-time  by  a  certain  portion  of  a  luminous 
object  and  the  corresponding  portion  of  the  image,  respectively. 

m 

154.  The  total  number  of  refracting  surfaces  of  a  system  is  denoted 
by  m\  also,  the  total  number  of  components  (prisms,  lenses  or  lens- 
combinations)  of  a  compound  optical  system. 

n,  n 

155.'  The  absolute  indices  of  refraction  of  the  first  and  second  medium 
are  denoted  by  n,  n'  ,  respectively. 

When  a  ray  traverses  a  series  of  media,  the  symbol  n'k  =  nk+l 
is  used  to  denote  the  absolute  index  of  refraction  of  the  (k  +  i)th 
medium.  Note  that  n'Q  =  nl  denotes  the  absolute  index  of  refraction 
of  the  first  medium. 

40 


610  Geometrical  Optics,  Appendix. 

Often,  also,  the  symbols  n,  ri  are  used  to  denote  the  absolute  indices 
of  refraction  of  the  first  and  last  medium,  respectively. 

The  symbols  n,  n  and  n  are  used  to  denote  the  absolute  indices  of 
refraction  of  a  medium  for  rays  of  light  of  wave-lengths  X,  X  and  I, 
respectively.  The  symbols  n,  n,  tt  and  n',  n',  n'  refer  to  the  first  and 
second  (or  to  the  first  and  last)  medium,  respectively. 

So,  also,  nA,  nB,  nc,  etc.  are  used  to  denote  the  absolute  indices  of 
refraction  of  a  medium  for  rays  of  light  corresponding  to  the  FRAUN- 
HOFER  lines  A,  B,  C,  etc.,  respectively. 

156.  In  an  optical  system  wherein  there  are  only  two  different 
media,  as,  for  example,  in  a  glass  lens  (or  prism)  surrounded  by  air, 
the  relative  index  of  refraction  from  the  first  medium  to  the  second  is 
usually  denoted  by  n\  thus,  n  =  njn^  =  n[/n'2.     In  this  sense,  the 
symbol  nk  is  regularly  employed  to  denote  the  index  of  refraction  of 
the  material  of  the  kih  lens  of  a  System  of  Infinitely  Thin  Lenses, 
each  of  which  is  surrounded  by  air. 

P,P 

157.  In  Chap.   XIII,  P,  P0  denote  the  "purity"  and  the  "ideal 
purity",  respectively,  of  the  spectrum.      In  this  chapter,  also,  the 
resolving  power  of  a  prism  or  prism-system  is  denoted  by  p. 

Q 

158.  The  invariant-functions  of  the  chief  ray  of  an  infinitely  narrow 
bundle  of  rays  refracted  at  a  spherical  surface  are  denoted  by  Q,  Q 
(or  (?A,  &);  see  Chap.  XII,  §299. 

r 

159.  The  function  T  =  h,.hk(Jk  —  JJ  denotes  a  certain  constant 
which  has  the  same  value  for  each  surface  of  a  centered  system  of 
spherical  surfaces;  see  Chap.  XII,  §323. 

V 

160.  In  Chap.  XIV,  the  symbol  V  is  used  to  denote  the  character- 
istic magnifying  power  of  an  optical  instrument  which  is  intended  to 
be  used  subjectively  in  conjunction  with  the  eye. 

161.  In  Chap.  XI,  §  247,  Vk  denotes  the  so-called  "constant  of  astig- 
matism" for  the  &th  surface  of  a  centered  system  of  spherical  surfaces. 

W 

162.  In  Chap.  XIV,  W  denotes  the  ratio  of  the  visual  angles  sub- 
tended at  the  eye,  on  the  one  hand,  by  the  image  as  viewed  through 


Symbols  of  Non -Geometrical  Magnitudes.  611 

the  instrument,  and,  on  the  other  hand,  by  the  object  as  seen  by  the 
naked  eye  at  the  distance  of  distinct  vision. 

x,  x,  X 

163.  In  an  Infinitely  Thin  Lens,  the  symbols  x,  xr  are  used  to 
denote  the  reciprocals  of    the  abscissae,  with  respect  to  the  optical 
centre  A  (2),  of  the  points  M,  M'  (28)  where  a  paraxial  ray  crosses 
the  axis  before  entering  the  lens  and  after  leaving  it,  respectively; 
thus,  x  =  i/u,  x'  =  i/uf  (104).     Similarly,  in  a  centered  system  of 
Infinitely  Thin  Lenses,  xk,  xk  denote  the  same  reciprocals  with  respect 
to  the  &th  lens. 

Similarly,  also,  the  symbols  x,  x  are  employed  as  follows  (see  104) : 

x  =  i/u,     *'  =  !/«';     xh  =  i/uk,      xk  =  i/<; 
x  =  i/u,     x'  =  i/u';     x'k  =  i/uk,     x'k  =  i/u'k. 

164.  The  so-called  "axial  magnification'1  (or  "depth  magnification") 
with  respect  to  a  pair  of  conjugate  axial  points  M,  M'  of  two  col- 
linear  systems  is  denoted  by  X;  thus,  if  FM  =  x,  E!M'  =  xf  (n,  12, 
24  and  107),  we  have:  X  —  dx'/dx. 

The  symbols  X,  X  denote  the  axial  magnifications  of  an  optical 
system  with  respect  to  a  given  axial  object-point  M  for  rays  of  light 
of  wave-lengths  X,  X,  respectively. 

In  Chap.  VII,  X0  is  used  to  denote  the  axial  magnification  at  the 
point  0  (see  35). 

F,  Y 

165.  F  denotes  the  so-called  "lateral  magnification"  at  a  pair  of 
conjugate  axial  points  M,  M'  (24)  of  two  collinear  systems:  F  =  y'/y. 
In  an  optical  system  composed  of  a  centered  system  of  m  spherical 
surfaces,    F  =  y^/y^     See  in. 

F,  F  denote  the  lateral  magnifications  of  an  optical  system  with 
respect  to  a  given  axial  object-point  M  for  rays  of  light  of  wave-lengths 
X,  X,  respectively. 

Y  denotes  the  lateral  magnification  at  the  pupil-centres  M,  M'. 

In  Chap.  VII,  F0  denotes  the  lateral  magnification  at  the  axial 
object-point  0  (see  35). 

166.  In  the  theory  of  the  refraction  of  an  infinitely  narrow  bundle 
of  rays,  F,  FM  denote  the  lateral  magnifications  of  the  collinear  plane- 
systems  TT,  TT'  and  TT,  ir',  respectively  (68). 


612  Geometrical  Optics,  Appendix. 

z,z 

167.  The  so-called  "angular  magnification"  (or  "convergence-ratio"} 
at  a  pair  of  conjugate  axial  points  M,  M'  (24)  of  two  collinear  systems 
is  denoted  by  Z;  Z  =  tan  0'  /tan  0  (129).     In  an  optical  system  com- 
posed of  a  centered  system  of  m  spherical  surfaces,  Z  =  tan  0^/tan  0^ 

Z,  Z  denote  the  angular  magnifications  of  an  optical  system  with 
respect  to  a  given  axial  object-point  M  for  rays  of  light  of  wave- 
lengths X,  X,  respectively. 

Z  denotes  the  angular  magnification  at  the  pupil-centres  M,  M'  . 

In  Chap.  VII,  Z0  denotes  the  angular  magnification  at  the  axial 
object-point  0  (see  35). 

168.  In  the  theory  of  the  refraction  of  an  infinitely  narrow  bundle 
of  rays,  Zu,  ZM  denote  the  convergence-ratios  of  the  meridian  and  sagittal 
rays,  respectively,  where  u  designates  the  chief  ray  of  the  bundle; 
Zu  =  dX'/^X,  Zu  =  d\fjd\  (134). 


169.  In  Chap.   XIII,  /3  is  used  to  denote  the  so-called  "relative 
partial  dispersion"  of  an  optical  medium.     For  the  &th  medium,  this 
magnitude  is  denoted  by  j8A. 

v 

170.  The  symbol  v  is  employed  to  denote  the  so-called  "relative 
dispersion"  of  an  optical  medium.     See  formulae  (366),  (426). 

<P 

171.  The  symbol  <p  is  used  to  denote  the  reciprocal  of  the  primary 
focal  length  of  an  Infinitely   Thin   Lens  —  the  so-called  "power"  or 
"strength"  of  the  lens. 

In  a  system  of  infinitely  thin  lenses,  <pk  denotes  the  power  of  the  &th 
lens,  and  <p  is  used  to  denote  the  power  of  the  Lens-System. 


INDEX. 


The  numbers  refer  to  the  pages. 


A. 


ABBE,  E.,  86,  104,  114,  159,  179,  201,  218, 
223,  233,  246,  262,  299,  342,  346,  349, 
353,  354,  374,  380,  382,  383,  385,  395,  397, 
398,  401,  405,  406,  407,  422,  434,  437,  448, 
450,  451,  467,  469,  478,  479,  480,  482,  485, 
492,  493,  509,  510,  523,  526,  528,  530, 
537,  538,  539,  543,  547,  548,  549,  557,  558, 
560,  563,  575,  578,  579;  theory  of  optical 
imagery,  198-201;  definitions  of  the  focal 
lengths,  233;  measure  of  the  indistinct- 
ness or  lack  of  detail  of  the  image,  385, 
395;  explanation  and  proof  of  the  sine- 
condition,  400-405-  use  of  term  "aplan- 
atic",  407;  test  of  aplanatism,  407,  422; 
method  of  invariants,  434,  448;  use  of 
term  "numerical  aperture",  538;  aboli- 
tion of  chromatic  difference  of  spherical 
aberration,  527;  use  of  the  terms  "pupils" 
and  "iris",  537;  optical  measuring  instru- 
ments ("telecentric"  systems),  541-544; 
definition  of  magnifying  power,  548;  in- 
vestigation of  focus-depth  and  accommo- 
dation-depth, 557-563;  and  of  illumina- 
tion in  the  Image-Space,  578. 

Aberration,  Chromatic:  see  Chromatic  Ab- 
errations, Achromatism,  etc.;  see  also 
Table  of  Contents,  Chap.  XIII. 

Aberration,  Lateral:  see  Lateral  Aberration, 
Spherical  Aberration. 

Aberration,  Least  Circle  of,  378. 

Aberration,  Longitudinal:  see  Spherical 
Aberration,  Chromatic  Aberration. 

Aberration-Curve,  398. 

Aberration-Lines:  comatic,  448-455;  of  as- 
tigmatic bundle  of  image-rays,  430-434, 
and  approximate  formulae  therefor,  432, 
440. 

Aberrations  or  Image-Defects,  368;  of  the 

'  third  order,  373;  sagittal  and  tangential 
(or  z-  and  y-aberrations) ,  374;  series- 
developments  of,  371-376,  397-400,  456- 
468. 

Aberrations,  Spherical:  see  Spherical  Aber- 
rations; also,  Table  of  Contents,  Chap. 
XII. 

Aberrations,  Theory  of:  see  Spherical  Aber- 
rations, Chromatic  Aberrations;  also  Table 
of  Contents,  Chaps.  XII  and  XIII. 


Abscissa,  Special  use  of  this  term,  52,  213, 
596. 

Absorption,  10,  12. 

Acanonical  system  of  co-ordinate  axes,  223. 

Accommodation-Depth,  561. 

Accommodation  of  the  eye,  563. 

Achromatic  Combinations,  Early  attempts 
at  contriving,  476. 

Achromatic  Combination  of  prisms,  489; 
of  two  thin  prisms,  483. 

Achromatic  Optical  System,  504. 

Achromatism,  476;  different  kinds  of,  503; 
complete  and  partial,  504;  stable,  507. 
See  Chromatic  Aberrations. 

Achromatism  with  respect  to  the  visual, 
and  with  respect  to  the  actinic  rays,  525. 

Affinity-relation  between  Object-Space  and 
Image-Space,  209,  243;  of  two  plane- 
fields,  206;  of  conjugate  planes  parallel 
to  the  focal  planes,  211. 

Affinity-relation  in  case  of  refraction  at  a 
plane  or  through  a  prism,  59,  71,  91,  100, 
123. 

AIRY,  Sir  G.  B.,  348,  420,  438;  his  tangent- 
condition,  420. 

ALHAZEN,  15. 

AMICI,  J.  B.,  491,  492,  502,  503;  direct- 
vision  prism-system,  492. 

ANDERSON,  A.,  127. 

ANDING,  E.,  573. 

Angles  of  incidence,  reflexion  and  refract- 
ion, 13;  angle  of  deviation,  27;  slope- 
angle,  135,  296,  316;  critical  angle  of 
refraction,  24. 

Angle-true  delineation,  418. 

Angular  Magnification  (Z):  see  Conver- 
gence-Ratio. 

Angular  Magnitudes,  Symbols  of,  604-608. 

Angular  (or  Inclined)  Mirrors,  54. 

Anomalous  Dispersion,  475. 

Aperture,  Numerical,  538. 

Aperture  of  Objective,  Choice  of  suitable, 
400. 

Aperture-angle,  538. 

Aperture-stop,  533. 

Aplanatic,  Meaning  of  the  term,  407. 

Aplanatic  Points  of  an  optical  system,  407; 
only  one  pair  of  such  points,  409. 

Aplanatic  Points  (Z,  Z')  of  refracting 
sphere,  290,  300,  346,  348,  387,  400,  405; 

613 


614 


Index. 


sine-condition  fulfilled  with  respect  to, 
291,  301,  401 ;  not  conjugate  points  in  the 
sense  of  collinear  imagery,  401;  comatic 
aberrations  vanish  for  this  pair  of  points, 
455- 

Aplanatism,  407;  ABBE'S  method  of  testing 
for,  407,  422. 

Apochromatic  Optical  System,  530. 

Apparent  Distance,  in  sense  used  by  COTES, 
192;  "apparent  distance"  of  object 
viewed  through  a  system  of  thin  lenses, 
191-197. 

Apparent  size  of  object,  544,  546;  of  image, 
546;  of  slit-image  as  seen  through  a 
prism  or  prism-system,  105. 

ARAGO,  D.  F.  J.,  19. 

Astigmatic  Bundle  of  Rays,  44-50;  merid- 
ian and  sagittal  sections  of,  46;  primary 
and  secondary  image-points  and  image- 
lines  of,  46.  See  Meridian  Rays  Sagittal 
Rays,  I  mage- Points,  Image-Lines,  Infi- 
nitely Narrow  Bundle  of  Rays,  Astigma- 
tism, etc. 

Astigmatic  Bundles  of  Rays,  Imagery  by 
means  of,  349-356,  402-405. 

Astigmatic  Constant,  357. 

Astigmatic  Difference,  in  case  of  narrow 
bundle  of  rays  refracted  (i)  at  a  plane,  60; 
(2)  through  a  prism,  94-97;  (3)  across  a 
slab,  1 08;  (4)  through  a  system  of  prisms, 
12 1 ;  (5)  at  a  spherical  surface,  345;  and 
(6)  through  a  centered  system  of  spheri- 
cal surfaces,  358. 

Astigmatic  Image-Surfaces,  416,  429,  430. 

Astigmatic  Refraction:  (i)  at  a  plane,  64- 
73.  360,  361;  (2)  through  a  prism,  90- 
IQ6;  (3)  across  a  slab,  106-111;  (4) 
through  a  prism-system,  115-123;  (5) 
at  a  spherical  surface  or  through  a  cen- 
tered system  of  spherical  surfaces,  see 
Table  of  Contents,  Chaps.  XI  and  XII; 
and  (6)  through  an  infinitely  thin  lens, 
363-366. 

Astigmatism,  STURM'S  Theory  of,  44-50; 
measure  of  the,  346;  historical  note  con- 
cerning, 347;  condition  of  the  abolition 
of  astigmatism  in  the  case  of  a  centered 
optical  system,  439.  See  Curvature  of 
Image;  see  also  Table  of  Contents, 
Chaps.  Ill,  IV,  XI  and  XII. 

Axes  of  co-ordinates  of  Object-Space  and 
Image-Space,  212;  positive  directions  of, 
220,  221,  227;  canonical  and  acanonical 
systems,  223. 

Axes,  Principal:  see  Principal  Axes. 

Axial  or  Depth-Magnification  (X),  234;  in 
case  of  telescopic  imagery,  244;  of  a 
centered  optical  system,  510;  chromatic 
variation  of,  511. 

Axis  of  collineation,  163. 

Axis  of  reflecting  or  refracting  sphere,  134; 
see  also  Optical  Axis. 


B. 

Barrel-shaped  distortion,  421. 

BARROW,  I.,  347. 

BECK,  A.,  199. 

BEER,  A.,  573. 

Bending  of  lens,  390. 

BESSEL,  F.  W.,  262,  263. 

BLAIR,  R.,  523. 

Blur-circle,  541,  555-560. 

Bow,  R.  H.,  421;  BOW-SUTTON  condition, 
421. 

BRANDES,  400. 

BRAVAIS,  31,  127,  128. 

BRETON  DE  CHAMP,  P.,  438. 

BREWSTER,  Sir  D.,  55. 

Brightness,  Definition  of,  579;  of  a  point- 
source,  581;  of  a  luminous  object,  579; 
of  optical  image,  580;  natural  brightness, 
580. 

BROWNE,  W.  R.,  406,  577. 

BRUNS,  H.,  38,  472. 

Bundle  of  rays,  41;  bundles  of  rays  and 
planes,  202;  homocentric  (or  monocen- 
tric)  bundle  of  rays,  44;  astigmatic  bun- 
dle of  rays,  44~5o;  general  characteristic 
of  infinitely  narrow  bundle  of  rays,  42-50. 
See  also  Astigmatic  Bundle  of  Rays,  In- 
finitely Narrow  Bundle  of  Rays. 

Bundle  of  Rays,  Character  of,  in  case  of 
direct  refraction  at  a  spherical  surface, 
376-380. 

Bundle  of  Rays,  Wide-angle,  necessary  for 
formation  of  image,  42,  287,  367. 

BUNSEN,  R.,  477. 

BURMESTER,  L.,  94,  97,  98,  99,  104,  III,  123, 

128,  336,  358;  homocentric  refraction 
through  prism  or  prism-system,  see  Table 
of  Contents,  Chap.  IV;  homocentric  re- 
fraction through  a  lens,  358. 

C. 

Calculation  of  the  path  of  a  ray  refracted 
at  a  spherical  surface,  (i)  in  a  principal 
section,  298,  299,  302;  (2)  not  in  a  prin- 
cipal section,  304-315.  See  A.  KERBER, 
L.  SEIDEL;  see  also  Table  of  Contents, 
Chap.  IX. 

Calculation  of  the  path  of  a  ray  refracted 
through  a  centered  system  of  spherical 
surfaces,  (i)  in  the  principal  section,  316— 
321;  numerical  illustration,  318-321;  (2) 
not  in  the  principal  section,  322-330. 
See  A.  KERBER,  L.  SEIDEL;  see  also 
Table  of  Contents,  Chap.  X. 

Camera,  Pin-hole,  288. 

Candle-power  of  point-source,  572. 

Canonical  system  of  axes  of  co-ordinates, 
223. 

Cardinal  points  of  optical  system,  179,  236. 

CAUCHY,  A.  L.,  474,  501. 

Caustic  Surfaces,  in  general,  42-44;  caustic 


Index. 


615 


curves,  43;  caustic  by  refraction  at  a 
plane,  59-64;  caustic  surfaces  in  case  of  a 
direct  bundle  of  rays  refracted  at  a 
sphere,  376. 

Centered  System  of  Spherical  Surfaces, 
Astigmatic  Refraction  of  narrow  bundle 
of  rays  through  a:  see  Table  of  Contents, 
Chaps.  XI  and  XII. 

Centered  System  of  Spherical  Surfaces, 
Calculation  of  the  path  of  a  ray  through 
a:  see  Table  of  Contents,  Chap.  X. 

Centered  System  of  Spherical  Surfaces,  Re- 
fraction of  paraxial  rays  through  a,  174— 
179;  law  of  R.  SMITH,  267;  formulae  of 
L.  SEIDEL,  269-273;  focal  lengths,  264-- 
267,  271;  focal  points  and  principal 
points,  177,  178,  271;  angular  magnifi- 
cation (Z)  and  axial  magnification  (X), 
510;  lateral  magnification  (F),  178,  510. 
See  Table  of  Contents,  Chaps.  VI  and 
VIII. 

Centered  System  of  Spherical  Surfaces, 
Spherical  and  Chromatic  Aberrations: 
see  Table  of  Contents,  Chaps.  XII  and 
XIII. 

Central  Collineation  of  two  plane-fields, 
162—173;  characteristics  of,  163;  project- 
ive  relations,  163;  geometrical  construct- 
ions, 165;  invariant  (c),  168;  character- 
istic equation,  170;  cases  that  occur  in 
Optics,  171-173. 

Central  perpendicular,  295. 

Centre  of  Collineation,  163. 

Centres  of  Perspective  (K  and  C)  of  ranges 
of  I.  and  II.  object-points  and  image- 
points  on  chief  rays  of  narrow  bundles 
of  incident  and  refracted  rays  in  case  of 
refraction  at  a  sphere,  339,  340,  343,  348; 
also  in  case  of  refraction  at  a  plane,  360, 
361 ;  and  of  reflexion  at  a  spherical  mirror, 
362,  363. 

Centres  of  perspective  of  Object-Space  and 
Image-Space,  540. 

Characteristic  Function  of  HAMILTON,  36- 
39- 

CHARLIER,  C.  V.  L.,  473. 

CHAULNES,  Due  de,  no. 

Chief  Ray,  as  representative  of  bundle  of 
rays,  41,  540;  defined  as  ray  that  goes 
through  the  centre  of  the  aperture-stop, 
32S.  375.  54o;  regarded  as  determining 
the  place  of  the  image-point  in  the  image- 

<     plane,  416,  540-544. 

CHRISTIE,  W.  H.  M.,  128. 

Chromatic  Aberration  of  a  system  of  thin 
lenses,  517-522;  of  two  thin  lenses  in  con- 
tact, 519;  of  two  separated  thin  lenses, 
520.  See  also  Chromatic  Variations. 

Chromatic  Aberrations,  Image  affected 
with,  503. 

Chromatic  Aberrations,  Theory  of:  see 
Table  of  Contents,  Chap.  XIII. 


Chromatic  Axial  or  Longitudinal  Aberra- 
tion of  a  centered  optical  system,  508. 

Chromatic  Dispersion,  475 ;  see  Dispersion. 

Chromatic  Under-  and  Over-Corrections, 
517. 

Chromatic  Variations  of  the  position  and 
size  of  the  image,  difference-formulae  of 
the,  505-510,  512;  differential  formulae 
of  the,  510,  511. 

Chromatic  Variations  of  the  focal  lengths, 
505.  foil. 

Chromatic  Variations  in  special  cases:  (i) 
single  lens  in  air,  513-516;  (2)  infinitely 
thin  lens,  516,  517.  See  also  Chromatic 
A  Serrations. 

Chromatic  Variations  of  the  Spherical  Aber- 
rations, 504,  526-531;  of  the  longitudinal 
aberration  along  the  axis,  526-530;  of 
the  sine-ratio,  526,  530. 

Circle  (or  Place)  of  Least  Confusion,  48, 
349,  429,  433. 

CLAIRAUT,  A.  C.,  477. 

CLASSEN,  J.,  262,  472. 

CLAUSIUS,  R.,  406,  577;  sine-condition, 
406. 

CODDINGTON,  H.,  348,  407,  438. 

Collinear  Imagery,  essentially  different 
from  "sine-condition"  imagery,  401,  408, 
411. 

Collinear  Optical  Systems,  218-262. 

Collinear    Plane-Fields,   162-173,  201-206. 

Collinear  relations  in  the  case  of  the  refract- 
ion of  a  narrow  bundle  of  rays  at  a 
spherical  surface,  35i~356,  402-405;  and 
through  a  centered  optical  system,  358- 
360. 

Collinear  Space-Systems,  162,  206-210; 
conjugate  planes  of,  210;  metric  rela- 
tions, 213-217;  lateral  magnifications, 
214. 

Collineation,  Central:  see  Central  Collinea- 
tion. 

Collineation,  Centre  and  Axis  of,  163. 

Collineation,  Definition  of,   162,  201,  foil. 

Collineation,  Theory  of,  as  applied  to 
Geometrical  Optics,  201-217;  see  Table 
of  Contents,  Chap.  VII;  see  also  Central 
Collineation. 

Colour  of  a  body  due  to  selective  absorp- 
tion, 10. 

Colour-phenomena:  see  Table  of  Contents, 
Chap.  XIII. 

Coma,  Origin  and  meaning  of  the  term,  445. 

Coma- Aberrations,  in  general,  444;  for- 
mulae for  the  comatic  aberration-lines  of 
the  meridian  rays,  448-455;  condition  of 
the  abolition  of  coma,  455;  comatic  aber- 
ration in  case  of  a  refracting  sphere,  455, 
and  of  an  infinitely  thin  lens,  455. 

Combination  of  two  or  more  optical  sys- 
tems, 245-262;  special  cases  of  combina- 
tions of  two  optical  systems,  251-255; 


616 


Index. 


focal  points  and  focal  lengths  of  com- 
pound systems,  245-250,  255-262. 

Compound  Optical  Systems:  see  Combina- 
tion of  optical  systems. 

Confusion,  Circle  of  least,  48,  349,  429,  433. 

Congruence  and  symmetry,  in  special  sense, 
223. 

Conjugate  Abscissae  of  projective  point- 
ranges,  213. 

Conjugate  Planes  of  Object-Space  and 
Image-Space,  210, 

Conjugate  Points  (or  Foci),  41;  construct- 
ions of  conjugate  points  of  optical  sys- 
tem, 241. 

Conjugate  Rays  of  Object-Space  and 
Image-Space,  Analytical  investigation  of, 
229. 

Convergence  of  the  meridian  and  sagittal 
rays  of  a  narrow  bundle  of  rays  re- 
fracted at  a  sphere,  Different  degrees  of, 
333- 

Convergence-Ratio  or  Angular  Magnifica- 
tion (Z),  234;  in  the  case  of  telescopic 
imagery,  245;  in  the  case  of  a  single 
spherical  refracting  surface,  264;  and  of 
a  centered  system  of  spherical  surfaces, 
510;  chromatic  variation  ofjjju. 

Convergence- Ratios  (Zu  and  ZM)  of  narrow 
pencils  of  meridian  and  sagittal  rays  re- 
/  fracted  (i)  at  a  plane,  68,  69;  (2)  through 
a  prism,  93,  94;  (3)  across  a  slab,  108; 
(4)  through  a  prism-system,  120;  and  (5) 
at  a  spherical  surface,  342,  345,  403-405. 

Convergent  and  Divergent  Optical  Systems, 
228. 

Co-ordinates,  Axes  of:  see  Axes  of  co-ordi- 
nates. 

CORNU,  A.,  31,  127,  339. 

Correction-terms  of  3rd  order,  in  Theory  of 
Spherical  Aberrations,  374-376,  458,  foil. 

COTES,  R.,  192,  193,  195,  198,  268;  formula 
for  the  "apparent  distance",  191-197. 

Critical  Angle  of  Refraction  (A)  with  re- 
spect to  two  media,  24. 

CROVA,  A.,  128. 

CULMANN,  P.,  192,  268,  348,  350,  353,  366. 

Curvature,  Lines  of,  of  a  surface,  43. 

Curvature  of  Image,  429—444;  development 
of  formulae  for  the  curvatures  of  the 
astigmatic  image-surfaces,  434,  441 ;  cur- 
vature of  the  stigmatic  image,  439;  cur- 
vature of  image  in  case  of  refracting 
sphere,  442 ;  and  of  an  infinitely  thin  lens, 
443.  See  also  Astigmatism. 

Cushion-shaped  distortion,  421. 

CZAPSKI,  S.,  48,  50,  114,  201,  217,  218,  233, 
246,  262,  335,  336,  349,  350,  353,  379,  400, 
448,  479,  480,  485,  492,  493,  523,  528, 
530,  547.  558,  560,  563,  578;  his  great 
work  on  the  theory  of  optical  instru- 
ments, 201;  arguments  in  favour  of  the 
image-lines  of  STURM,  50,  335;  imagery 


by  means  of  astigmatic  bundles  of  rays, 
349- 

D. 

D'ALEMBERT,  J.,  477. 

Depth-Magnification :  see  Axial  Magnifica- 
tion. 

Depth  of  Accommodation,  561. 

Depth  of  Focus:  see  Focus-Depth. 

Depth  of  Vision,  562. 

DESCARTES,  R.,  15. 

Deviation  of  refracted  ray,  27;  deviation 
of  ray  refracted  through  a  prism  (i)  in 
principal  section,  78-81;  (2)  obliquely, 
125;  through  a  prism-system,  113,  114. 
See  also  Minimum  Deviation. 

Deviation  without  dispersion  in  a  prism- 
system,  489;  in  a  combination  of  two  thin 
prisms,  483. 

Diagrams,  Designations  of  points,  lines  and 
surfaces  in  the,  583-596. 

Diagrams  for  showing  procedures  of  paraxial 
rays,  142. 

Diffraction-effects,  4;  diffraction-pattern  as 
image,  42. 

Direction  of  ray  or  straight  line:  see  Posi- 
tive Direction. 

Direct-vision  prism-system,  491;  of  AMICI, 
492,  502;  combination  of  two  thin  prisms, 
484. 

Dispersion,  Analytical  formula  for,  ^485; 
anomalous,  475;  characteristic  dispersion 
of  a  medium,  485;  chromatic  dispersion, 
475;  irrationality  of  dispersion,  482; 
mean  dispersion,  481;  partial  dispersion, 
481,  523;  relative  dispersion  (  =  i/*7),  481; 
relative  partial  dispersion  (/3) ,  523;  resi- 
dual dispersion,  523. 

Dispersion  in  case  of  (i)  a  single  prism  in 
air,  487;  (2)  a  system  of  prisms,  484-492; 
especially,  a  train  of  prisms  composed 
alternately  of  glass  and  air,  488. 

Dispersion,  Minimum,  of  a  prism,  487. 

Dispersion  without  deviation:  see  Direct- 
vision  prism-system. 

Dispersive  strength  and  "dispersor"  of  lens, 
516. 

Distinct  Vision,  Distance  of,  547. 

Distortion  of  image,  and  conditions  of  its 
abolition,  415-429,  467;  measure  of  the 
distortion,  417;  typical  kinds  of  distor- 
tion, 421:  distortion  in  case  the  pupil- 
centres  are  the  pair  of  aplanatic  points, 
421.  See  also  Orthoscopy. 

Distortion-aberration,  Development  of  the 
formula  for,  422-427;  in  the  case  of  a 
refracting  sphere,  427,  and  of  an  infinitely 
thin  lens,  428. 

DlTSCHEINER,  L.,   128. 

Divergent    and    Convergent    Optical  Sys- 
tems, 228. 
DOLLOND,  J.,  476,  477. 


Index. 


617 


DONDERS,   562. 

DRUDE,  P.,  21,  407,  571. 

E. 

Effective  Rays,  41,  537. 

Electromagnetic  Theory  of  Light,  2. 

Emission  Theory  of  Light,  i,  19. 

Entrance-port,  550;  optical  system  with 
two  entrance-ports,  551;  optical  project- 
ion-system with  one  entrance-port,  563- 
568;  and  with  two  entrance-ports,  568- 

571- 

Entrance-pupil,  323,  374,  537;  reciprocity 
between  object  and  entrance-pupil,  537. 

Equivalent  Light-Source,  574. 

EULER,  L.,  36,  42,  370,  476,  477. 

EUCLID,  15. 

EVERETT,  J.  D.,  407. 

Exit-port,  550;  see  Entrance-port. 

Exit-pupil,  375,  537;  reciprocity  between 
image  and  exit-pupil,  537-  See  also  En- 
trance-pupil. 

F. 

FERMAT,  P.,  33,  34.  4°,  4995  law  of,  33. 

Field  of  view,  549-551;  angular  measure  of, 
550,  551;  in  a  projection-system  of  finite 
aperture  (i)  with  one  entrance-port,  563- 
568;  and  (2)  with  two  entrance-ports 
568-571- 

Field-stop,  550. 

FlNSTERWALDER,  S.,  471. 

Flat  (or  plane)  image,  Conditions  of,  440, 
467.  See  Curvature  of  Image. 

"Flucht"  Lines  of  conjugate  planes,  204; 
see  Focal  Lines. 

"Flucht"  Planes  of  two  collinear  space- 
systems,  208;  see  Focal  Planes. 

"Flucht"  Points  of  projective  point-ranges 
(or  of  conjugate  rays),  166,  203,  219; 
see  also  Focal  Points. 

Fluorescence,  12. 

Focal  Lengths  (/,  e'}  of  Object-Space  and 
Image-Space,  Definitions  of,  233,  237; 
also  157;  relations  to  the  image-constants 
a,  b,  233. 

Focal  Lengths  (/,  e')  of  centered  system  of 
spherical  surfaces,  264-267,  271;  of  com- 
pound optical  system,  248,  258;  of  spher- 
ical refracting  surface,  155,  264;  of  thick 
lens,  275;  of  thin  lens,  284;  of  infinitely 
thin  lens,  186,  284;  of  a  system  of  two 
lenses,  285.  Focal  Length  (/)  of  spher- 

'    ical  mirror,  140. 

Focal  Lengths  (/«,  eu'  and  fu,  eu')  of  the 
meridian  and  sagittal  collinear  systems 
of  a  narrow  bundle  of  rays  refracted  at  a 
spherical  surface,  354;  and  through  a 
centered  system  of  spherical  surfaces,  359. 

Focal  Lines,  154,  168,  206;  see  "Flucht" 
Lines. 

Focal  Planes  (<£,  e')  of  Object-Space  and 


Image -Space,  208;  of  refracting  sphere, 
154.     See  also  "Flucht"  Planes. 

Focal  Planes,  Sine-Condition  in  the,  408. 

Focal  Points  (F,  E')  of  Object-Space  and 
Image-Space,  211;  of  centered  system  of 
spherical  surfaces,  177,  271 ;  of  compound 
optical  system,  247,  256;  of  refracting 
sphere,  150;  of  spherical  mirror,  140;  of 
thick  lens,  275;  of  infinitely  thin  lens,  184, 
284;  of  a  system  of  two^  lenses,  285. 

Focal  Points  (/,  V  and  T,  I")  of  'meridian 
and  sagittal  rays  of  narrow  bundle  of 
rays  in  case  of  refraction  at  a  spherical 
surface,  340-344;  in  case  of  refraction  at 
a  plane,  361;  in  case  of  reflexion  at  a 
spherical  mirror,  362;  SMITH'S  construct- 
ion of  the  Focal  Points  J,  I',  348. 

Focal  Surface,  471. 

Foci,  Conjugate,  41;  focus,  44. 

Focus-Depth  of  projection-systems  of  finite 
aperture,  557-560;  of  systems  of  finite 
aperture  used  in  conjunction  with  the  eye, 
560,  561;  lack  of  detail  in  the  image  due 
to,  560. 

Focus-Plane,  552. 

FOUCAULT,  L.,  19. 

FRAUNHOFER,  J.,  21,  83,  87,  104,  319,  415, 
469,  470,  475,  477,  478,  479,  482,  502,  525, 
526,  528,  529,  610;  so-called  FRAUN- 
HOFER-Condition,  essentially  same  as 
Sine-Condition  and  condition  of  abolition 
of  Coma,  415;  FRAUNHOFER-Lines  of  the 
solar  spectrum,  477,  478. 

FRESNEL,  A.,  i,  7,  9;  explanation  of  the 
so-called  rectilinear  propagation  of  light, 
7,8. 

G. 

GALILEO,  i,  398,  478. 

GAUSS,  C.  F.,  54,  178,  179,  198,  199,  .200, 
233,  237,  239,  263,  319,  367,  369,  371,  372, 
373,  374,  379,  380,  385,  400,  401,  403,  416, 
417,  422,  430,  431,  432,  433,  440,  446,  454, 
456,  457,  458,  459.  464,  5o6,  526,  528, 
532,  534.  588,  590,  591,  595,  603;  his 
famous  work  on  Optics,  198;  his  defini- 
tions of  the  focal  lengths,  233 ;  use  of  the 
Principal  Points,  237;  GAUSsian  Imagery, 
263,  367,  369;  GAUSsian  parameters 
of  the  incident  and  refracted  rays,  456- 
459;  so-called  GAUSS'S  Condition,  528. 

GEHLER,  G.  S.  T.,  128,  400. 

Geometrical  Optics,  Its  scope  and  plan,  2,3. 

Geometrical  Theory  of  Optical  Imagery: 
see  Table  of  Contents,  Chap.  VII. 

Glass,  Optical:  Kinds  of,  480;  Jena  Glass, 
478-483:  investigations  of  ABBE  and 
SCHOTT,  478. 

GLEICHEN,  A.,  97,  98,  114,  358. 

GOERZ,  P.,  358. 

Graphical  Method  of  showing  imagery  by 
Paraxial  Rays,  142. 


618 


Index. 


GRUBB,  T.,  127. 
GRUNERT,  J.  A.,  239. 

GUENTHER,  S.f  262. 

H. 

HALL,  476. 

HAMILTON,  Sir  W.  R.,  33,  36,  38,  39,  472; 
HAMILTON'S  Characteristic  Function,  36- 
39.  472. 

HANKEL,  H.,  199. 

HARTING,  H.,  366. 

HEATH,  R.  S.,  40,  44,  55,  125,  349,  476. 

HELMHOLTZ,  H.  VON,  44,  98,  128,  192,  197, 
268,  353,  406,  493,  494,  497,  577;  so- 
called  HELMHOLTZ  Equation,  197,  268 
(see  also  SMITH,  LAGRANGE);  proof  of 
Sine-Condition,  406,  577;  measure  of 
purity  of  spectrum,  493. 

HEPPERGER,  J.  VON,  128. 

HERO  of  Alexandria,  33. 

HERSCHEL,  J.  F.  W.,  394,  407, 470,  487,  556; 
HERSCHEL-Condition,  394,  470,  556. 

HERTZ,  H.,  2. 

HOCKIN,  C.,  407. 

Homocentric  Bundle  of  Rays,  Definition, 
44. 

Homocentric  Image-Points,  with  respect 
to  prism,  1 02,  133;  with  respect  to  slab, 
109-111;  with  respect  to  prism-system, 
122;  with  respect  to  lens,  358,  364. 

Homocentric  Refraction  through  a  prism, 
97-105,  128-133;  through  a  prism-sys- 
tem, 122;  across  a  slab,  loo-m;  through 
a  lens,  358. 

HOORWEG,  J.  L.,  127,  128. 

HUYGENS,  C.,  i,  3,  4,  5,  6,  7,  16,  17,  18,  522; 
construction  of  wave-front  in  general,  4- 
7;  construction  of  reflected  and  refracted 
wave-fronts,  16-19;  HUYGENS'S  Ocular, 
522. 

I. 

Illumination,  Intensity  of,  571-575. 

Illumination  in  the  Image-Space,  578. 

Image,  ABBE'S  measure  of  the  indistinctness 
or  lack  of  detail  of  the,  385;  numerical 
calculation,  395-397. 

Image,  Diffraction-,  42. 

Image,  Flat,  Conditions  of,  440,  467. 

Image  of  extended  object  by  astigmatic 
bundles  of  rays,  349-351;  see  also  Astig- 
matism. 

Image,  Order  of,  according  to  PETZVAL,  370, 
37i. 

Image,  Perfect  or  Ideal,  41,  198;  MAX- 
WELL'S definition,  200. 

Image,  Practical,  367-369,  412;  require- 
ments of  a  good  image,  367;  practical 
images  by  means  of  wide-angle  bundles 
of  rays,  412;  and  by  means  of  narrow 
bundles,  416,  417. 

Image,  Projected  on  screen-plane,  553,  555. 


Image,  Real  or  Virtual,  41;  erect  or  in- 
verted, 228. 

Image-Constants  (a,  b,  c),  Signs  of  the,  218, 
228:  their  connections  with  the  focal 
lengths  (/,  e'),  232. 

Image-Equations:  In  general,  216;  in  case 
of  symmetry  around  the  principal  axes, 
222;  in  terms  of  the  focal  lengths,  233; 
referred  to  conjugate  axial  points,  and, 
especially,  to  the  Principal  Points  (A,  A'), 
239,  240;  in  the  case  of  Telescopic  Im- 
agery, 243,  244,  245;  in  the  case  of  a  re- 
fracting sphere,  158-162,  264;  in  the  case 
of  an  infinitely  thin  lens,  187. 

Image-Lines,  Primary  and  Secondary,  of 
narrow  astigmatic  bundle  of  image-rays, 
44-50,  331,  334-336,  348;  directions  of, 
48-50,  335;  exactly  what  is  meant  by, 
48-50,  335- 

Image-Lines,  Primary  and  Secondary,  of 
narrow  astigmatic  bundle  of  rays  re- 
fracted at  a  plane,  65,  66;  through  a 
prism,  94;  at  a  spherical  surface,  331, 
334-336,  348. 

Image-Plane,  370. 

Image-Point  or  Point-Image,  according  to 
Physical  Optics,  42,  287,  368. 

Image-Points,  Primary  and  Secondary,  of 
narrow  astigmatic  bundle  of  image-rays, 
46,  331 ;  see  also  Infinitely  Narrow  Bundle 
of  Rays. 

Image-Points,  Primary  and  Secondary,  of 
narrow  astigmatic  bundle  of  rays  re- 
fracted (i)  at  a  plane,  65-68,  360;  con- 
structions of  I.  Image-Point,  71,  361;  (2) 
through  a  prism:  construction  of,  90; 
formulae  for,  92;  (3)  across  a  slab:  con- 
struction of,  1 06;  formulae  for,  107;  (4) 
through  a  prism-system :  construction  of, 
115;  formulae  for,  117;  (5)  at  a  spherical 
surface,  336-348;  (6)  through  a  centered 
system  of  spherical  surfaces,  356-358; 
(7)  in  case  of  reflexion  at  a  spherical 
mirror,  362. 

Image-Points,  Homocentric,  102;  see  Homo- 
centric  Image-Points. 

Imagery,  Ideal,  by  means  of  Paraxial  Rays: 
see  Table  of  Contents,  Chap.  VIII;  see 
also  Optical  Imagery,  GAUSSICM  Imagery. 

Imagery  in  the  planes  of  the  meridian  and 
sagittal  sections  of  an  infinitely  narrow 
bundle  of  rays  refracted  at  a  spherical 
surface,  349-356,  402-405. 

Imagery,  So-called  SEIDEL-:  see  L.  SEIDEL. 

Image-Space  and  Object-Space:  see  Object- 
Space  and  Image-Space. 

Image-Surfaces,  Astigmatic,  416,  429,  430; 
curvatures  of,  434-441;  see  Curvature  of 
Image. 

Incidence,  Angle  of,  13;  incidence-height, 
135.  295;  incidence-normal,  13;  plane  of 
incidence,  13;  incidence-point,  13. 


Index. 


619 


Incident  light,  10;  incident  ray,  13. 

Inclined  Mirrors,  54,  55. 

Index  of  Refraction:  Absolute,  20;  relative, 
14,  20;  "artificial"  index  of  refraction  in 
case  of  oblique  refraction,  30,  126;  con- 
nection between  index  of  refraction  and 
wave-length  of  light,  474. 

Indistinctness  (or  Lack  of  Detail)  of  image 
due  to  spherical  aberration  along  the  axis, 
ABBE'S  measure  of,  385,  395-397- 

Infinitely  Narrow  Bundle  of  Rays,  42-50, 
331-336;  see  Table  of  Contents,  Chaps. 
II,  III,  IV  and  XI;  also  Chap.  XII  under 
Astigmatism;  see  also  Astigmatic  Refract- 
ion. 

Infinitely  Narrow  Bundle  of  Rays,  Astig- 
matic Reflexion  of,  at  a  spherical  mirror, 
361-363. 

Infinitely  Narrow  Bundle  of  Rays,  Astig- 
matic Refraction  of,  at  a  spherical  surface 
or  through  a  centered  system  of  spherical 
surfaces,  see  Table  of  Contents,  Chaps.  XI 
and  XII;  collinear  correspondence,  351- 
356,402-405;  lateral  magnifications  (Yu, 
FM)  of  the  meridian  and  sagittal  rays, 
403-405;  imagery  in  the  planes  of  the 
meridian  and  sagittal  sections,  349-356, 
402-405. 

Infinitely  Thin  Lens,  Chromatic  Variations 
in  case  of  an,  516. 

Infinitely  Thin  Lens,  Refraction  of  Paraxial 
Rays  through  an,  173,  182-190,  388;  a 
case  of  central  collineation,  173;  conju- 
gate axial  points,  183;  focal  points,  184, 
284;  focal  lengths,  186,  284;  power  or 
strength  (0),  186;  dispersive  strength 
and  "dispersor",  516;  lateral  magnifica- 
tion (F),  187;  imagery,  187. 

Infinitely  Thin  Lens,  Special  Notation  in 
case  of  an,  387. 

Infinitely  Thin  Lens,  The  Spherical  Aber- 
rations in  case  of  an:  Astigmatism,  363- 
366;  Axial  or  Longitudinal  Aberration, 
387-392;  Coma,  455;  Curvature  of  Image, 
443;  Distortion- Aberration,  428. 

Infinitely  Thin  Lenses,  System  of:  Chro- 
matic Aberration  of  a,  517,  foil. 

Infinitely  Thin  Lenses,  System  of:  Refract- 
ion of  Paraxial  Rays  through  a,  190; 
lateral  magnification  (F),  191;  COTES'S 
formula  for  "apparent  distance"  of  object 
viewed  through  such  a  system,  191-197. 
System  of  two  infinitely  thin  lenses,  285. 
System  of  infinitely  thin  lenses  in  contact, 
191. 

Infinitely  Thin  Lenses,  System  of:  Spheri- 
cal (Longitudinal)  Aberration  of  a,  392- 
394;  case  when  lenses  are  in  contact, 

393- 

Initial  Values  of  the  Ray- Parameters:  in 

KERBER'S   Refraction-Formulae,    323;  in 

SEIDEL'S    Refraction-Formulas,     329;  in 


SEIDEL'S  Theory  of  Spherical  Aberra- 
tions, 373-375- 

Interval  (A)  between  two  consecutive  com- 
ponents of  a  compound  optical  system, 
247;  in  case  of  a  lens,  274. 

Invariant  (c)  of  Central  Collineation,  168; 
invariant  (7)  of  refraction  at  a  sphere, 
299;  invariant  (J)  of  refraction  of  par- 
axial  rays  at  a  sphere,  so-called  "zero 
invariant",  159;  invariant,  optical  (K  -- 
n  -sin  a),  21. 

Invariant-Method  of  E.  ABBE,  especially 
as  applied  to  the  development  of  the 
formulae  for  the  curvatures  of  the  image- 
surfaces,  434,  foil.;  and  of  the  formulae 
for  the  comatic  aberrations,  448,  foil. 

Invariants  (Q,  (?)  of  Astigmatic  Refraction, 
434,  450. 

Iris  of  optical  system,  537. 

Isoplethic  Curves  of  VON  ROHR,  529. 

J. 

Jena  glass,  478-483 ;  table  of  some  varieties 

of,  480;  optical  properties  of,  482. 
JETTMAR,  H.  VON,  128. 

K. 

KAESTNER,  A.  G.,  191. 

KAYSER,  H.,  79, 114,  128,  485,  488,  493,  498. 

KEPLER,  15. 

KERBER,  A.,  305,  306,  310,  311,  312,  322, 
467,  473,  528,  529,  602,  605,  606,  607; 
Refraction-Formulae,  305-307,  310-312, 
322-325;  chromatic  correction  of  optical 
system,  528. 

KESSLER,  F.,  73,  79,  288,  336,  360,  514; 
investigations  of  the  chromatic  aberra- 
tions of  a  lens,  514. 

KIRCHHOFF,  G.  R.,  192,  477,  577. 

KlRKBY,  J.  H.,   79. 

KLEIN,  F.,  38. 

KLINGENSTIERNA,  S.,  476. 

KOENIG,  A.,  312,  376,  398,  412,  420,  422, 
43«.  434.  446,  448,  455.  468,  473,  506,  509, 
512,  513,  519.  523.  524.  526,  528. 

KOHLRAUSCH,  F.,  86. 

KRUESS,  H.,  526,  528. 

KUMMER,  E.  E.,  44,  48,  335,  351,  471. 

KUNDT,  A.,  21. 

KURZ,  A.,  79. 

L. 

LAGRANGE,  J.  L.  DE,  192, 195,  197,  268,  353; 
so-called  LAGRANGE-HELMHOLTZ  Equa- 
tion, 197,  268;  see  HELMHOLTZ,  SMITH. 

LAMBERT,  J.  H.,  538,  573. 

Lateral  Aberration,  379;  formula  for,  385; 
chromatic,  510. 

Lateral  Magnification  (F)  of  two  collinear 
space-systems,  214-216,  221,  234;  rela- 
tion to  the  other  magnification-ratios 


620 


Index. 


(X  and  Z),  234;  in  the  case  of  a  telescopic 
system,  244. 

Lateral  Magnification  (F)  in  case  of  a 
spherical  mirror,  145. 

Lateral  Magnification  ( F)  in  case  of  refract- 
ion at  a  plane,  58;  at  a  sphere,  160,  264; 
through  a  centered  system  of  spherical 
surfaces,  178,  510;  through  a  lens,  180; 
through  an  infinitely  thin  lens,  187; 
through  a  system  of  thin  lenses,  191. 

Lateral  Magnification  (F),  Chromatic 
variation  of,  505,  foil. 

Lateral  Magnifications  (Fu,  FM)  of  the 
meridian  and  sagittal  sections  of  narrow 
astigmatic  bundle  of  rays  refracted  at  a 
spherical  surface  (or  through  a  centered 
system  of  spherical  surfaces),  403-405. 

Law  of  independence  of  rays  of  light,  2,  8, 
9;  of  rectilinear  propagation  of  light  2, 
3  -8;  of  reflexion  and  of  refraction,  2,13-20. 

Least  Action,  MAUPERIUIS'S  Principle  of, 
36. 

Least  Circle  of  Aberration,  378. 

Least  Confusion,  Place  or  Circle  of,  48,  349, 

429.  433. 

Least  Time,  Principle  of,  33-36. 

Left-screw  imagery,  223,  225,  228. 

Lens,  Definition,  179;  types  of  lenses,  180; 
convergent  and  divergent,  positive  and 
negative  lenses,  180,  185;  character  of 
the  different  forms  of  lenses,  276-283; 
bending  of  lenses  390;  rectilinear  lens, 
418.  See  also  Thick  Lenses,  Thin  Lenses, 
Infinitely  Thin  Lenses,  Lens-Systems. 

Lens-Systems,  284-286;  system  of  two 
lenses,  284;  two  systems  of  lenses,  285; 
system  of  two  infinitely  thin  lenses, 
285.  See  also  Infinitely  Thin  Lenses. 

LEONARDI  DA  VINCI,  i. 

L'HospiTAL,  348. 

LIE,  S.,  472. 

Light,  Mode  of  propagation  of,  2,  4;  see 
also  Rectilinear  Propagation  of  Light. 

Light,  Theories  of:  Emission,  i,  19;  Wave, 
i,  19;  Electromagnetic,  2. 

Light,  Tube  of,  571. 

Light,  Velocity  of,  19. 

Light-rays:  see  Rays  of  Light. 

Light-Source,  Equivalent,  574. 

Linear  Magnitudes,  Symbols  of,  596-604. 

Lines,  Designations  of,  in  the  diagrams,  593, 
594- 

LIPPICH,  F.,  44,  199,  239,  262,  288,  332,  339, 
352,  354.  36o. 

LISTING,  J.  B.,  238. 

LOEWE,  F.,  98,  99,  114,  128,  485,  492,  493. 

LOMMEL,  E.,  79. 

Longitudinal  Aberration:  see  Spherical  Ab- 
erration and  Chromatic  Aberration. 

LUCAS  of  Liege,  482. 

Luminous  Surface-Element,  Radiation  of, 
573- 


LUMMER,  O.,  41,  349,  368,  408,  471,  527. 


M. 


Magnification,  Angular  (Z):  see  Conver- 
gence-Ratio. 

Magnification,  Axial  or  Depth  (X):  see 
Axial  Magnification. 

Magnification,  Lateral  (F):  see  Lateral 
Magnification. 

Magnification,  Objective,  544. 

Magnification- Ratio,  214;  relation  of  the 
magnification-ratios  to  each  other,  234; 
in  the  case  of  telescopic  imagery,  243-245. 

Magnifications,  Different,  of  the  different 
zones  of  a  spherically  corrected  system, 
402. 

Magnifying  Power,  Objective,  544;  subject- 
ive, 545-549;  intrinsic  magnifying  power 
of  optical  system,  548. 

MALUS,  E.  L.,  39,  40,  42;  law  of  MALUS,  39, 
40. 

MATTHIESSEN,  L.,  48,  49,  50,  64,  239,  262, 

335.  351- 

MAUPERTUIS,  36. 

MAXWELL,  J.  C.,  2,  38,  199,  200,  201; 
theory  of  perfect  optical  instruments, 
199,  200. 

Measurement,  Optical  Instruments  for  pur- 
pose of,  541. 

Medium,  Optical,  9;  transparent,  translu- 
cent, opaque,  13. 

Meridian  Planes  of  Optical  System,  212, 
214,  227. 

Meridian  Rays  of  narrow  bundle  of  rays: 
see  Astigmatism,  Astigmatic  Refraction, 
etc.;  see  also  Table  of  Contents,  Chaps. 
II,  III,  IV,  XI  and  XII. 

Meridian  Rays  of  narrow  bundle  of  rays 
refracted  at  a  sphere,  331;  convergence 
of,  333;  collinear  relations,  351-356;  lat- 
eral magnification  (Fu),  403;  imagery, 

353.  403- 

Meridian  Section  of  narrow  astigmatic 
bundle  of  rays,  46,  331. 

Meridian  Section  of  bundle  of  rays  refracted 
at  a  sphere,  Lack  of  symmetry  in  the, 
445,  446-448. 

Metric  Relations  of  two  collinear  space- 
systems,  213-217. 

MICHELSON,  A.  A.,  19. 

Micrometer-Microscope,  542. 

Minimum  Deviation  by  a  prism,  78,  79, 
81-83,  87,  99,  127;  in  case  of  oblique 
refraction  through  prism,  126;  minimum 
deviation  by  a  prism-system,  114. 

Minimum  Dispersion  by  a  prism,  487. 

Minimum  Property  of  the  Light-Path:  see 
Least  Time:  see  also  FERMAT,  HAMILTON. 

MINOR,  01. 

Mirror:  see  Plane  Mirror,  Spherical  Mirror. 

MOEBIUS,  A.  F.,  44,   178,   199,   201,   262. 


Index. 


621 


Monocentric  Bundle  of  Rays,   Definition, 

44. 

MONOYER,  F.,  262. 

MUELLER,  FR.  C.  G.,  79- 

MUELLER-POUILLET'S  Lehrbuch  der  Physik, 
349,  368,  407,  408,  527. 

N. 

NEUMANN,  C.,  44. 

NEUMANN,  C.  G.,  239. 

NEWTON,  Sir  I.,  i,  19,  192,  347.  348,  475, 
476,  478,  482;  discoverer  of  astigmatism, 
347;  prism-experiments,  475. 

Nodal  Points  (N,  N')  and  Planes  of  optical 
system,  238;  nodal  points  of  refracting 
sphere,  264;  of  lens,  275. 

Normal  Sections  of  a  surface,  28,  42. 

Notation,  System  of:  see  Appendix,  583- 
612. 

Numerical  Aperture,  538. 

Numerical  Illustration  of  the  calculation  of 
the  path  of  a  ray  through  an  optical  sys- 
tem: (i)  Paraxial  Ray,  320;  and  (2) 
Edge- Ray  in  principal  section,  321;  of 
the  calculation  of  the  spherical  aberra- 
tion, 319-321,  394-397- 

O. 

Object,  Projected  on  focus-plane,  553,  555 

Objective  Magnification,  and  Magnifying 
Power,  544. 

Object-Plane,  370. 

Object-Space  and  Image-Space,  207;  geo- 
metrical characteristics  of ,  210-212;  con- 
jugate planes  of,  210;  focal  points  (F,  £'), 
21 1 ;  principal  axes  (x,  #')»  212;  axes  of 
co-ordinates,  212;  positive  directions  of 
co-ordinate  axes,  220,  221,  227;  relation 
between  conjugate  rays,  229;  focal  lengths 
(/,  «'),  232. 

Oblique  Refraction  in  general,  28-32;  con- 
struction of  obliquely  refracted  ray,  31. 

Oblique  Refraction  at  a  plane,  311,  315. 

Oblique  Refraction  at  a  sphere:  Parameters 
of  the  ray,  304-310;  KERBER'S  Refraction- 
Formulae,  310-312;  SEIDEL'S  Refraction- 
Formulae,  313-315- 

Oblique  Refraction  through  a  centered 
system  of  spherical  surfaces:  KERBER'S 
Refraction-Formulae,  322-325;  SEIDEL'S 
Refraction-Formulae,  325-330.  .  • 

Oblique  Refraction  through  a  prism:  Con- 

;  struction  of  the  path  of  the  ray,  123;  cal- 
culation of  the  path,  124;  deviation  of 
the  ray,  125.  Oblique  Refraction  of 
narrow  bundle  of  rays  through  a  prism, 
128-133. 

Oculars  of  HUYGENS  and  RAMSDEN,  522. 

Optical  Axis  of  spherical  surface,  134;  of 
centered  system,  174,  227;  positive  direc- 
tion of,  135,  227. 

Optical  Centre  of  Lens,  181. 


Optical  Image,  40,  42;  from  standpoint  of 
Physical  Optics,  42,  287,  368;  brightness 
of,  579-582;  intensity  of  radiation  of, 
575-579- 

Optical  Imagery,  Geometrical  Theory  of: 
see  Table  of  Contents,  Chap.  VII;  ABBE'S 
Theory  of,  198-201;  characteristic  metric 
relation  of,  213,  220;  general  character- 
istics of,  218-229;  different  types  of ,  223- 
229.  See  also  Imagery. 

Optical  Instrument,  Function  of,  in  general, 
198;  MAXWELL'S  definition  of  "perfect" 
optical  instrument,  200. 

Optical  Invariant  (K  =  n  •  sin  a),  21. 

Optical  Length,  35. 

Optical  Measuring  Instruments,  541. 

Optical  Systems,  Collinear,  218-262;  com- 
bination of,  245-262;  convergent  and 
divergent,  228. 

Order  of  image,  according  to  PETZVAL,  370, 
371- 

Orthoscopic  (or  Angle- true)  image,  418. 

Orthoscopic  Points  of  Optical  System,  421. 

Orthoscopy,  415—429;  condition  of,  in  gen- 
eral, 418;  in  case  the  system  is  spherically 
corrected  with  respect  to  the  pupil-cen- 
tres, 420.  See  Distortion. 

Orthotomic  system  of  rays,  40. 

P. 

Parallel  Plane  Refracting  Surfaces,  Path  of 
ray  traversing  a  series  of,  89 ;  see  also  Slab. 

Parameters  of  incident  and  refracted  rays 
in  case  of  refraction  at  a  spherical  surface, 
296;  in  case  of  oblique  refraction  at  a 
spherical  surface,  304-310;  initial  values 
in  case  of  oblique  refraction  through  a 
centered  system  of  spherical  surfaces, 
323-329;  parameters  of  GAUSS,  456-459; 
parameters  used  by  SEIDEL  in  his  theory 
of  spherical  aberrations,  371,  459;  ap- 
proximate values  of  the  SEiDEL-param- 
eters  and  the  corrections  of  the  3rd 
order,  372,  459,  foil.;  relations  of  the 
SEiDEL-parameters  to  those  of  GAUSS, 
459- 

Paraxial  Ray,  Definition,  136;  numerical 
illustration  of  calculation  of  path  of  par- 
axial  ray  through  a  centered  system  of 
spherical  surfaces,  320. 

Paraxial  Rays,  Ideal  Imagery  by  means 
of:  see  Table  of  Contents,  Chap.  VIII; 
graphical  method  of  showing  imagery 
by  paraxial  rays,  142. 

Paraxial  Rays,  Reflexion  and  Refraction  of, 
at  a  spherical  surface:  see  Table  of  Con- 
tents, Chap.  V. 

Paraxial  Rays,  Refraction  of,  at  a  plane,  57- 
59,  161,  172;  through  a  centered  system 
of  spherical  surfaces  or  through  a  lens  or 
lens-system:  see  Table  of  Contents, 
Chaps.  VI  and  VIII. 


622 


Index. 


Path  of  ray  reflected  at  a  spherical  mirror, 
299. 

Path  of  ray  refracted  at  a  plane,  55-56,  300, 
3".  315. 

Path  of  ray  refracted  at  a  spherical  surface : 
Calculation  of  (i)  when  ray  lies  in  a  prin- 
cipal section,  298,  299,  302;  and  (2)  when 
ray  does  not  lie  in  a  principal  section, 
304-315;  see  Table  of  Contents,  Chap. 
IX.  Geometrical  investigation  of  path 
of  ray  refracted  at  spherical  surface,  288- 
294;  YOUNG'S  construction  of  refracted 
ray,  288. 

Path  of  ray  refracted  through  a  centered 
system  of  spherical  surfaces:  (i)  when 
ray  lies  in  a  principal  section,  316-321; 
and  (2)  when  ray  does  not  lie  in  a  princi- 
pal section,  322-330.  See  Table  of  Con- 
tents, Chap.  X. 

PEACOCK,  G.,  339. 

Pencil  of  rays,  41,  202. 

Perspective,  Centres  of:  see  Centres  of  Per- 
spective. 

Perspective  elongation  of  projected  object 
(or  image),  553. 

PETZVAL,  J.,  370,  371,  438,  439,  440,  529, 
564;  order  of  the  image,  370,  371;  for- 
mula for  curvature  of  image,  439;  field 
of  view  of  projection-system,  564. 

PEZENAS,  LE  PERE,  191. 

Photograph,  Correct  distance  of  viewing  a, 
554- 

Pin-hole  camera,  288. 

Plane-field,  Definition,  163,  202;  central 
collineation  of  two  plane-fields,  162-173; 
plane-fields  in  collinear  relation,  201-206; 
projective  relation,  163,  202;  affinity-re- 
lation of  plane-fields,  206. 

Plane  Image,  Conditions  of,  440,  467;  see 
Curvature  of  Image, 

Plane  Mirror,  51-55;  conjugate  points  with 
respect  to,  51;  collinear  imagery,  52,  288; 
image  of  extended  objects  in  a,  53;  uses 
of,  54;  number  of  images  by  successive 
reflexions  in  a  pair  of  plane  mirrors,  54, 
55- 

Plane  Surface,  Path  of  ray  refracted  at  a, 
55.  56,  300;  KERBER'S  formulae  for  path 
of  ray  refracted  obliquely  at  a,  311;  and 
SEIDEL'S  formulae  for  the  same,  315. 

Plane  Surface,  Reflexion  at  a:  see  Plane 
Mirror. 

Plane  Surface,  Refraction  of  paraxial  rays 
at  a,  55,  56,  161,  172. 

Plane  Surface,  Astigmatic  Refraction  of 
narrow  bundle  of  rays  at  a,  64-73,  360, 
361;  geometrical  relations  between  ob- 
ject-points and  image-points,  70;  con- 
struction of  the  I.  Image-Point,  71;  and 
of  the  I.  and  II.  Image- Points,  360,  361. 

POGGENDORF,  J.   C.,  54. 

Point-Image:  see  Image-Point. 


Point-Range,  Definition,  202;  projectively 
similar  ranges  of  points,  211;  directly  and 
oppositely  projective  point-ranges,  219. 
See  Projective  Relations,  Affinity-Rela- 
tions. 

Points,  Designations  of,  in  the  diagrams, 
583-593. 

Point-Source  of  light,  6;  radiation  of,  571; 
candle  power  of,  572;  brightness  of,  581. 

Ports,  Entrance-  and  Exit-,  549;  see  En- 
trance-Port. 

Positive  Directions  of  incident,  reflected 
and  refracted  rays,  22,  251;  positive 
direction  of  optical  ray,  219;  of  a  straight 
line,  22;  of  the  principal  axes  (*,  *')  of 
the  Object-Space  and  Image-Space,  220, 
221,  227;  of  the  secondary  axes  of  co- 
ordinates, 221,  227;  of  the  optical  axis, 
135.  227. 

Power,  or  Strength  (0),  of  infinitely  thin 
lens,  1 86. 

Primary  and  Secondary  Astigmatic  Image- 
Surfaces,  416,  429,  430;  curvatures  of, 
434-441.  See  Curvature  of  Image. 

Primary  and  Secondary  Focal  Lengths:  see 
Focal  Lengths. 

Primary  and  Secondary  Focal  Points:  see 
Focal  Points. 

Primary  and  Secondary  Image-Points  and 
Image-Lines:  see  Image-Points,  Image- 
Lines,  Infinitely  Narrow  Bundles  of  Rays, 
Astigmatism,  etc. 

Principal  Axes  (x,  #')  of  Object-Space  and 
Image-Space,  212;  positive  directions  of, 
220,  221;  symmetry  around,  221. 

Principal  Axes  (u,  uf)  of  the  two  pairs  of 
collinear  plane-systems  (TT,  irf  and  TT,  v') 
in  the  case  of  refraction  of  narrow  bundle 
of  rays  at  a  spherical  surface,  351-354. 

Principal  Planes  of  optical  system,  178,  237. 

Principal  Points  (A,  A')  of  optical  system 
in  general,  237;  image-equations  referred 
to,  239,  240. 

Principal  Points  (A,  A')  of  a  centered  sys- 
tem of  spherical  surfaces,  178,  271;  of  a 
spherical  refracting  surface,  264;  of  a 
lens,  275;  of  a  system  of  two  lenses,  285. 

Principal  Sections  of  a  surface,  42. 

Principal  Sections  of  a  prism,  74;  of  a  re- 
fracting sphere,  294. 

Prism,  Definition,  74;  refracting  angle,  74; 
principal  section,  74;  construction  of  ray 
refracted  through  prism  in  principal  sect- 
ion, 74;  total  reflexion  at  second  face  of 
prism,  78,  84,  87;  normal  emergence  at 
second  face,  83,  87;  deviation  of  ray  by 
prism,  78-81,  125;  ray  of  minimum  de- 
viation, 78,  79,  81-83,  87,  99,  126,  127; 
path  of  ray  refracted  obliquely  through  a 
prism,  123-128. 

Prism,  Astigmatic  Refraction  of  narrow 
bundle  of  rays  through  a,  90-97. 


Index. 


623 


Prism,  Homocentric  Refraction  through  a, 
(i)  when  chief  ray  lies  in  a  principal 
section,  97-105;  (2)  when  chief  ray  is 
obliquely  refracted  through  prism,  128- 

133- 

Prism,  Dispersion  of,  487;  see  Dispersion. 
Prism-formulae,  Collection  of,  87. 
Prism-spectra:  see  Table  of  Contents,  Chap. 

XIII, 

Prism-System,  Achromatic,  483,  489-491. 
Prism-System,  Direct- vision,  484,  491,  502. 
Prism-System,  Dispersion  of,  484-492;  see 

Dispersion. 
Prism-System,  Path  of  Ray  through  a,  m- 

115;  construction,  112;  calculation,  113; 

condition   of    minimum   deviation,    114. 

See  Table  of  Contents,  Chap.  IV. 
Prism-System,    Resolving   power   of,    498- 

503- 

Prism,  Thin,  83;  achromatic  combination 
and  direct-vision  combination  of  two  thin 
prisms,  483,  484. 

Projected  Object  and  Image,  553;  in  case 
of  projection-systems  of  finite  aperture, 
555- 

Projections  of  incident  and  refracted  rays, 
Theorems  concerning,  30. 

Projective  Relation  of  two  collinear  plane- 
fields,  202;  in  special  case  of  Central 
Collineation,  163. 

Projective  Relations  of  ranges  of  I.  and  II. 
object-points  and  image-points  lying  on 
chief  rays  of  narrow  bundles  of  incident 
and  refracted  rays,  in  case  of  refraction 
at  a  spherical  surface,  338,  343,  345,  347. 

PTOLEIVLEUS,  C.,  15. 

PULFRICH,  C.,  86. 

Pupils,  Entrance-  and  Exit-,  532-540;  see 
Entrance-Pupil,  Exit-Pupil.  Planes  of 
the  Pupils,  374. 

Purity  of  the  Spectrum,  492-498;  Ideal 
Purity  of  Spectrum,  497;  see  Spectrum. 

R. 

RADAU,  R.,  75. 

Radiation  of  point-source,  571 ;  of  luminous 
surface-element,  573. 

RAMSDEN-ocular,  522. 

Range  of  points:  see  Point-Range. 

Rays  of  light,  2,  3,  4,  8-20;  mutual  inde- 
pendence of,  2,  8,  9;  rays  meet  wave- 
surface  normally,  39.  See  also  Bundle  of 

'    Rays,  Pencil  of  Rays,  etc. 

Rays,  Geometrical,  202 ;  orthotomic  system 
of  rays,  40. 

Rays,  Chief:  see  Chief  Rays. 

Ray-co-ordinates,  56,  296. 

Ray-length,  295. 

Ray-parameters:  see  Parameters  of  Ray. 

Ray- Path,  Reversibility  of,  15,  207.  See 
also  Path  of  Ray. 

RAYLEIGH,  Lord,  191,  192,  195,  268,  342, 


497,  498,  500,  502,  581;  resolving  power 
of  prism-system,  498-503. 

Real  and  virtual,  10;  real  and  virtual 
images,  41. 

Rectilinear  Propagation  of  Light,  2,  3-8. 

Reflected  Light,  10. 

Reflected  Ray,  13,  construction  of,  25;  de- 
viation of,  27;  positive  direction  of,  22, 
251- 

Reflexion,  Angle  of,  13;  laws  of,  2,  13-20; 
total,  22-25;  regular  and  irregular  (or 
diffused),  n;  reflexion  as  special  case  of 
refraction,  22,  161,  299,  361. 

Refracted  Light,  10. 

Refracted  Ray,  13;  construction  of,  26; 
deviation  of,  27;  positive  direction  of,  22. 

Refracting  Angle  of  Prism,  74. 

Refraction,  Angle  of,  13;  laws  of,  2,  13-20; 
regular  and  irregular,  n,  12;  index  of, 
see  Index  of  Refraction. 

Refraction-Formulae  of  A.  KERBER  and  L. 
SEIDEL:  see  KERBER,  SEIDEL;  see  also 
Table  of  Contents,  Chaps.  IX  and  X. 

Refraction  (or  Reflexion),  Oblique:  see 
Oblique  Refraction. 

Refractive  Index:  see  Index  of  Refraction. 

Residual  Dispersion:  see  Secondary  Spec- 
trum. 

Resolving  Power:  Of  eye,  369,  557,  559; 
of  prism-system,  498-503. 

REUSCH,  E.,  31,  72,  75,  91,  127,  142. 

Reversibility  of  Ray-Path,  Principle  of,  15, 
207. 

Right-screw  imagery,  223,  225,  228. 

ROETHIG,  O.,  262,  458. 

RUDOLPH,  P.,  529. 

S. 

Sagittal  or  z-aberrations,  374. 

Sagittal  Rays  of  narrow  bundle  of  rays:  see 
Astigmatism,  Astigmatic  Refraction,  etc.; 
see  also  Table  of  Contents,  Chaps.  II, 
III,  IV,  XI  and  XII. 

Sagittal  Rays  of  narrow  bundle  of  rays  re- 
fracted at  a  sphere,  333,  343-345;  con- 
vergence of,  334;  collinear  relations,  351- 
356;  lateral  magnification  (FM),  403-405; 
imagery,  353,  404;  symmetry  in  the  sag- 
ittal section,  445. 

Sagittal  Section  of  narrow  bundle  of  rays, 
46. 

SALMON,  G.,  60. 

SCHELLBACH,  K.,  288. 
SCHELLBACH,  R.  H.,  79. 
SCHLEIERMACHER,  L.,  37O. 

SCHMIDT,  W.,  475. 

SCHOTT,  O.,  87,  478,  479. 

SCHUSTER,  A.,  501. 

Screen-plane,  553. 

Secondary  Focal  Point,  Focal  Length, 
Image-Point,  Image-Line,  Image-Sur- 
face: see  Focal  Points,  Focal  Lengths, 


624 


Index. 


Image-Points,  Image-Lines,  Image-Sur- 
faces. 

Secondary  Spectrum,  479,  482,  504,  523- 
526;  of  a  system  of  thin  lenses  in  contact, 
524- 

SEIDEL,  L.,  268,  269,  270,  271,  297,  305,  307, 
308,  309,  310,  313,  314,  325,  326,  327,  328, 
329.  330,  366,  369,  370,  371,  373,  376,  415, 
438,  439,  440,  456,  458,  459,  465,  467,  468, 
469,  470,  471,  472,  510,  526,  607,  608;  for- 
mulae for  refraction  of  paraxial  rays 
through  a  centered  system,  269-273; 
formulae  for  calculation  of  path  of  ray 
through  centered  optical  system,  305, 
307-310,  313-315.  325-330;  SEIDEL 
Imagery,  369-376;  parameters  of  inci- 
dent and  refracted  rays,  307-310,  371- 
373.  4595  theory  of  the  spherical  aberra- 
tions of  the  3rd  order,  369-376,  456- 
473;  development  of  formulae  for  the  y- 
and  z-aberrations,  456-468;  SEIDEL'S 
Five  Sums,  467. 

Series-Developments  of  the  spherical  aber- 
rations of  the  3rd  order,  371-376,  397- 
400,  456-470. 

Series-Developments  of  the  comatic  aber- 
rations, 446,  448,  foil. 

Sheaf  of  Planes,  Definition,  202. 

Shortest  Route,  Principle  of  the,  35. 

Signs  of  the  Image-Constants  a,  b,  c,  218; 
228;  of  the  focal  lengths/,  e',  233. 

Similar  Ranges  of  Points,  71,  92,  99,  117; 
protectively  similar  ranges  of  points,  211, 
214,  243. 

Similarity  between  object  and  image,  Con- 
dition of,  222;  also,  418.  See  Distortion, 
Orthoscopy. 

SIMMS,  W.,  128. 

Sine-Condition,  400-415;  its  derivation  and 
meaning,  400-407;  derived  from  a  general 
law  of  R.  CLAUSIUS,  406;  development 
of  formulae  for  the  sine-condition,  412- 
415;  identical  with  condition  of  abolition 
of  Coma,  455;  sine-condition  in  the  focal 
planes,  408.  See  also  Aplanatism. 

Sine-Condition,  Chromatic  Variation  of, 
530. 

Sine-Condition  satisfied  with  respect  to 
aplanatic  points  (Z,  Z')  of  refracting 
sphere,  291,  301,  401. 

Slab,  with  plane  parallel  faces:  as  special 
case  of  prism,  86;  construction  of  path  of 
ray  across  a,  88;  astigmatic  refraction  of 
narrow  bundle  of  rays  across  a,  106-111; 
homocentric  refraction  across  a,  109-111. 

Slit-image,  as  seen  through  prism,  94,  105; 
as  seen  through  prism-system,  120;  dif- 
fraction-image of  slit,  495. 

Slope  of  ray,  135,  296,  316. 

SMITH,  R.,  191,  192,  195,  196,  197,  267,  268, 
347,  348,  353,  385,  401,  427,  440,  441,  461, 
465,  581;  his  law,  196,  197,  267;  proof  of 


COTES'S  formula  and  corollaries  there- 
from, 192-197;  construction  of  focal 
points  J,  I'  of  narrow  bundle  of  rays  in 
case  of  refraction  at  a  sphere,  348. 

SMITH-HELMHOLTZ  Equation,  268. 

SNELL,  W.,  15,  19,  158. 

SOUTHALL,  J.  P.   C.,   173,  2 1 8. 

Space-Systems,  in  collinear  relation,  206- 
210. 

Specific  Intensity  of  Radiation  of  luminous 
surface,  573. 

Spectrum,  475;  continuous  spectrum,  478; 
solar  spectrum,  478;  purity  of,  492-498; 
purity  of  spectrum  of  single  prism,  494; 
ideal  purity  of  spectrum,  497. 

Spherical  Aberration  in  its  narrow  sense,  so- 
called  Longitudinal  Aberration  or  Aber- 
ration measured  along  the  optical  axis, 
292,  300,  376-400,  467;  development  of 
general  formulae  for,  380-385 ;  formula  for 
the  abolition  of,  384,  467;  in  the  special 
cases  (i)  of  a  single  spherical  surface,  386; 
(2)  of  an  infinitely  thin  lens,  387-392;  (3) 
of  a  system  of  two  or  more  thin  lenses, 
392-394;  (4)  of  a  system  of  two  thin 
lenses  in  contact,  393.  The  terms  of  the 
higher  orders  in  the  series-developments, 
397-400. 

Spherical  Aberrations,  Theory  of:  see  Table 
of  Contents,  Chap.  XII;  see  also  Astig- 
matism, Coma,  Curvature  of  Image,  Dis- 
tortion-Aberration, Orthoscopy,  Sine-Con- 
dition, SEIDEL,  etc. 

Spherical  Aberrations,  Chromatic  Varia- 
tions of  the,  504,  526-531;  chromatic 
variation  of  the  longitudinal  aberration, 
526;  chromatic  variation  of  the  sine-con- 
dition, 530. 

Spherical  Mirror,  Astigmatic  Reflexion  of 
narrow  bundle  of  rays  at  a,  361-363. 

Spherical  Mirror,  Reflexion  of  paraxial  rays 
at  a,  137-147,  161;  a  case  of  central  col- 
lineation,  173;  conjugate  axial  points 
(M,  M')  with  respect  to,  137-142;  con- 
struction of  axial  image-point  M',  139; 
focal  point  and  focal  length,  140;  extra- 
axial  conjugate  points  (Q,  Q')  and  the 
lateral  magnification  (Y),  142-147;  con- 
struction of  image-point  Q',  144. 

Spherical  Mirror,  Path  of  ray  reflected  at  a, 
299. 

Spherical  Over- Correction  and  Under-Cor- 
rection, 378. 

Spherical  Refracting  Surface,  Aplanatic 
Points  of:  see  Aplanatic  Points. 

Spherical  Refracting  Surface,  Astigmatic 
Refraction  of  narrow  bundle  of  rays  at  a: 
see  Table  of  Contents,  especially  Chap. 
XL 

Spherical  Refracting  Surface,  Path  of  ray 
refracted  at  a:  Construction  of  the  ray, 
288;  trigonometric  formulae  for  calculat- 


Index. 


625 


-  ing  path  of  ray:  see  Table  of  Contents, 
Chap.  IX;  see  also  KERBER  and  SEIDEL. 

Spherical  Refracting  Surface,  Refraction  of 
Paraxial  Rays  at  a,  147-162,  264;  a  case 
of  Central  Collineation,  171;  conjugate 
axial  points  (M,  M'),  147-152;  construc- 
tion of  axial  image-point  M',  149;  the 
focal  points  (F,  £').  150;  construction  of 
extra-axial  image-point  Q',  153;  the  focal 
planes,  154;  the  focal  lengths  (/,  e"),  155, 
264;  the  image-equations,  158-162,  264; 
the  zero-invariant  (/),  159;  the  lateral 
magnification  (F),  160,  264;  the  angular 
magnification  (Z),  264;  principal  points 
and  nodal  points,  264. 

Spherical  Refracting  Surface,  Spherical  Ab- 
errations incase  of  a:  Astigmatism,  442; 
Comatic  Aberration,  455;  Curvature  of 
Image,  442;  Distortion- Aberration,  427; 
Spherical  or  Longitudinal  Aberration, 
380-383,  386. 

Spherical  Surfaces,  Centered  System  of: 
see  Centered  System  of  Spherical  Surfaces; 
see  also  Table  of  Contents,  Chaps.  VI, 
VIII,  X,  XI,  XII,  XIII  and  XIV. 

Spherically  Corrected  Optical  System,  377; 
different  magnifications  of  the  different 
zones  of,  402. 

STEINHEIL,  A.,  307,  313,  327,  329,  398. 

STEINHEIL,  R.,  473. 

Stigmatic  Image,  Curvature  of,  439. 

STOKES,  Sir  G.  G.,  127. 

Stops,  Effect  of,  532;  aperture-stop,  533; 
field-stop,  550;  front  and  rear  stops,  533; 
interior  stop,  374,  416,  420,  533;  virtual 
stop,  536. 

Strength  or  Power  (<£)  of  infinitely  thin  lens, 
186. 

STRUTT,  J.  W.:  see  RAYLEIGH. 

STURM,  J.  C.,  44,  46,  48,  50,  66,  335;  his 
theory  of  astigmatism,  44-50.  335. 

Subjective  Magnifying  Power,  545-549. 

Surfaces,  Designations  of,  in  the  diagrams, 
594-596. 

SUTTON,  T.,  421;  BOW-SUTTON  Condition, 
421. 

Symbols  of  angular  magnitudes,  604-608; 
of  linear  magnitudes,  596-604;  of  non- 
geometrical  magnitudes  (constants,  co- 
efficients, etc.),  608-612. 

Symmetry  around  principal  axes,  221. 

Symmetry    and     Congruence,     in    special 

•  senses,  223. 

Symmetry,  Lack  of,  of  a  pencil  of  meridian 
rays  of  finite  aperture,  445,  446-448. 

System  of  Lenses  and  System  of  Prisms: 
see  Lens-System,  Prism- System. 

T. 

TAIT,  P.  G.,  36,  571. 
Tangential  or  y-aberrations,  374. 


Tangent  Condition  of  AIRY,  420 ;  see  Orthos- 
copy. 

TAYLOR,  H.  D.,  319,  396,  438,  445. 

Telecentric  Optical  System,  543. 

Telescopes,  Oculars  of,  522. 

Telescopic  Imagery,  210,  243-245;  image- 
equations,  243,  244,  245;  characteristics 
of,  244. 

Telescopic  Optical  System,  162,  243-245; 
produced  by  a  combination  of  two  non- 
telescopic  systems,  251;  combination  of 
two  telescopic  systems,  254;  combination 
of  telescopic  with  non- telescopic  system, 
252. 

Thick  Lens,  Chromatic  Variations  in  case 
of  a,  513-516. 

Thick  Lens,  Refraction  of  Paraxial  Rays 
through  a,  179,  273-283;  lateral  magni- 
fication (F),  1 80;  optical  centre,  181; 
focal  points,  principal  points  and  focal 
lengths,  275.  See  Lenses,  Lens-Systems. 

Thicknesses  measured  along  the  optical 
axis,  177,  316;  thickness  of  a  lens,  179, 
274. 

THIESEN,  M.,  38,  472. 

Thin  Lens,  283;  see  Infinitely  Thin  Lens. 

Thin  Prism:  see  Prism. 

THOLLON,  L.,  488,  495. 

THOMPSON,  S.  P.,  41,  445,  471. 

TOEPLER,  A.,  238. 

Total  Reflexion,  22-25;  at  second  face  of 
prism,  78,  84. 

V. 

VEILLON,  H.,  79. 

Vertex  of  spherical  surface,  134. 

Vignette-angle,  565. 

Virtual  and  real,  10;  virtual  image,  41. 

Virtual  stop,  536. 

VOIT,  E.,  307,  313,  327,  329,  398. 

VON  HOEGH,  E.,  522. 

VON  ROHR,  M.,  98,  114,  128,  192,  218,  223, 
262,  268,  312,  348,  350,  353,  366,  376,  398, 
412,  420,  421,  422,  430,  434,  439,  446,  448, 
455,  468,  473,  479,  485,  492,  493,  506,  509, 
512,  513,  519,  523,  526,  528,  529,  536,  550, 
55L  554.  565. 

W. 

WADSWORTH,  F.  L.  O.,  498. 
WAGNER,  R.,  238. 
WANACH,  B.,  310,  366. 
WANDERSLEB,  E.,  218,  223,  262. 
Wave-Front,  HUGYENS'S  construction  of,  4- 

7;  in  special  cases,  17-19. 
Wave-Front,  met  by  rays  orthogonally,  39- 
Wave-length  of  light  and  refractive  index, 

474- 

Wave-motion,  i. 
Wave  Theory  of  Light,  I,  19. 
WEIERSTRASS,  288. 
WHEWELL,  W.,  200. 


626 


Index. 


Wide-angle     bundles    of    rays,     Practical  339, 348,  352,  360,  593,  596;  construction 


Images  by  means  of,  412. 
WILSING,  J.,  98,  128. 

WOLLASTON,  W.  H.,  477. 

WOOD,  R.  W.,  44- 

Y. 


of  ray  refracted  at  a  sphere,   288;  con- 
tributions to  theory  of  astigmatism,  348. 

Z. 

Zero-invariant  (J),  159. 


YOUNG,  T.,  i,  288,  289,  290,  292,  294,  336,       ZINKEN  gen.  SOMMER,  H.,  438,  440. 


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